AN IMPROVED DESIGN METHODOLOGY

FOR MODELING THICK-SECTION COMPOSITE

STRUCTURES USING A MULTI-SCALE APPROACH

by

Jeffrey Staniszewski

A thesis submitted to the

Faculty of the University of Delaware in partial fulfillment of the requirements

for the degree of Master of Science in Mechanical Engineering

Summer 2010

Copyright 2010 Jeffrey Staniszewski

All Rights Reserved

AN IMPROVED DESIGN METHODOLOGY

FOR MODELING THICK-SECTION COMPOSITE

STRUCTURES USING A MULTI-SCALE APPROACH

by

Jeffrey Staniszewski

Approved: __________________________________________________________

Michael Keefe, Ph.D.

Professor in charge of thesis on behalf of the Advisory Committee

Approved: __________________________________________________________

Anette M. Karlsson, Ph.D.

Chair of the Department of Mechanical Engineering

Approved: __________________________________________________________

Michael J. Chajes, Ph.D.

Dean of the College of Engineering

Approved: __________________________________________________________

Debra Hess Norris, M.S.

Vice Provost for Graduate and Professional Education

iii

ACKNOWLEDGMENTS

First, I would like to thank my family for their support and encouragement

throughout my degree program. I would like to thank my advisor, Dr. Michael Keefe,

for the guidance and expertise he has provided me over the past two years. I wish to

thank Dr. Travis Bogetti for guidance as well as encouraging me to continue my

education and giving me the opportunity to further my career. I would also like to

thank him for his continued support of me as a contractor at the Army Research

Laboratory throughout my graduate studies.

I wish to thank Dr. Brian Powers for his input and expertise on the

modeling and programming work that was needed to complete my research. I would

like to acknowledge the staffs of the Maryland Office of Oak Ridge Associated

Universities, the Weapons and Materials Research Division of the Army Research

Laboratory, and the Center for Composite Materials for their assistance throughout my

graduate studies.

The research was supported in part by an appointment to the Postgraduate

Research Participation Program at the U.S. Army Research Laboratory administered

by the Oak Ridge Institute for Science and Education through an interagency

agreement between the U.S. Department of Energy and USARL.

iv

TABLE OF CONTENTS

LIST OF TABLES ......................................................................................................... vi

LIST OF FIGURES ...................................................................................................... vii

ABSTRACT ................................................................................................................. xii

INTRODUCTION .......................................................................................................... 1

1.1 Problem statement ..................................................................................... 2

1.2 Benchmarking ............................................................................................ 3

1.2.1 State-of-the-art commercial composite models ............................. 3

1.2.2 State-of-the-art for other software products .................................. 5

1.2.3 Literature review............................................................................ 7

1.2.4 Selection of FEA software ........................................................... 10

SOLID MECHANICS APPROACH ............................................................................ 11

2.1 Smearing-unsmearing approach .............................................................. 11

2.2 Background of the “LAM” codes ............................................................ 12

2.3 Laminated media methodology ............................................................... 16

2.4 Nonlinear constitutive relationships ........................................................ 20

2.5 Incremental approach to predicting nonlinear laminate response ........... 22

2.6 Progressive ply failure methodology ....................................................... 24

IMPLEMENTATION ................................................................................................... 28

3.1 Theoretical consideration ........................................................................ 29

3.1.1 Ply-level stress and strain calculations ........................................ 29

3.1.2 Poisson’s ratio ............................................................................. 35

3.2 Procedural considerations ........................................................................ 36

3.2.1 Progressive failure adaption for FEA .......................................... 36

3.2.2 Tracking progressive failure ........................................................ 41

3.2.3 Selection of elements ................................................................... 42

3.3 Validation of LAMPATNL ..................................................................... 45

3.3.1 Validation of LAMPATNL for linear materials .......................... 47

3.3.2 Validation of LAMPATNL for nonlinear materials .................... 50

3.4 Validation summary ................................................................................ 64

CASE STUDIES........................................................................................................... 65

4.1 LAMPATNL output parameters .............................................................. 65

4.2 Design methodology ................................................................................ 70

4.3 Original LAMPATNL design methodology ............................................ 72

4.4 Case study #1 ........................................................................................... 78

4.4.1 Description .................................................................................. 78

v

4.4.2 Results of original design ............................................................ 80

4.4.3 Design iteration #1 ...................................................................... 93

4.4.4 Design iteration #2 .................................................................... 100

4.4.5 Design iteration #3 .................................................................... 101

4.5 Case study #2 ......................................................................................... 108

4.5.1 Description ................................................................................ 108

4.5.2 Results of original design .......................................................... 110

4.5.3 Design iteration #1 .................................................................... 123

4.5.4 Design iteration #2 .................................................................... 126

CONCLUSIONS ........................................................................................................ 134

5.1 Conclusions ........................................................................................... 134

5.2 Summary of results ................................................................................ 136

5.3 Future work............................................................................................ 137

REFERENCES ........................................................................................................... 140

REPRINT PERMISSION LETTER ........................................................................... 143

vi

LIST OF TABLES

Table 1.1 Percentage of keyword papers containing FEA software ........................ 10

Table 2.1 Failure mode and property modification ................................................. 26

Table 3.1 Examples provided for LAMPATNL validation ..................................... 47

Table 3.2 Linear ply properties of S-Glass/Epoxy ................................................... 48

Table 3.3 Ply properties for linear S-Glass/Epoxy .................................................. 48

Table 3.4 Ply properties for linear T300/PR-319 .................................................... 50

Table 3.5 Ramberg-Osgood parameters .................................................................. 52

Table 3.6 Maximum strain failure allowables ......................................................... 52

Table 4.1 List of state dependent parameters .......................................................... 66

vii

LIST OF FIGURES

Figure 2.1 LAMPAT process methodology ............................................................. 15

Figure 2.2 Laminate configuration [1] ..................................................................... 16

Figure 2.3 Relation of global coordinates to laminate and ply coordinates ............. 18

Figure 2.4 Nonlinear stress-strain response ............................................................. 21

Figure 2.5 Incremental laminate loading methodology [2] ...................................... 22

Figure 2.6 Effect of increment size on nonlinear stress-strain response .................. 24

Figure 3.1 Ply stress and strain calculation procedure for LAM3DNLP ................. 31

Figure 3.2 Ply stress and strain calculation procedure for LAMPATNL ................ 34

Figure 3.3 LAM3DNLP progressive failure procedure ........................................... 37

Figure 3.4 Stress-strain curves of [0°/90°/±45°]s S2 Glass/Epoxy laminate ........... 38

Figure 3.5 LAMPATNL progressive failure procedure ........................................... 39

Figure 3.6 Stress-strain curves of [0°/90°/±45°]s S2-Epoxy laminate -

LAM3DNLP output and ABAQUS UMAT output ............................... 40

Figure 3.7 Tracking failure history is mode, ply, integration point, and

element specific ...................................................................................... 42

Figure 3.8 Comparison of visualization of full integration (left) and reduced

integration (right) elements .................................................................... 44

Figure 3.9 Stress vs. strain curves in the X-direction for [0°/90°/±45°]s S-

Glass/Epoxy comparing the LAM3D and LAMPATNL codes.............. 49

Figure 3.10 Stress vs. strain curves in the Y-direction for [±30°]s T300/PR-

319 comparing the LAM3D and LAMPATNL codes ............................ 50

viii

Figure 3.11 Stress vs. strain curves in the 1-direction for [0°] S-Glass/Epoxy

comparing the UMAT and LAM3DNLP codes ..................................... 54

Figure 3.12 Stress vs. strain curves in the 2-direction for [0°] S-Glass/Epoxy

comparing the UMAT and LAM3DNLP codes ..................................... 55

Figure 3.13 Stress vs. strain curves in the 3-direction for [0°] S-Glass/Epoxy

comparing the UMAT and LAM3DNLP codes ..................................... 56

Figure 3.14 Stress vs. strain curves in the X-direction for [0°/90°/±45°]s S-

Glass/Epoxy comparing the UMAT and LAM3DNLP codes ................ 57

Figure 3.15 Stress vs. strain curves in the X-direction for [±35°]s IM7/8551-7

comparing the UMAT and LAM3DNLP codes ..................................... 58

Figure 3.16 Stress vs. strain curves in the Y-direction for [±35°]s IM7/8551-7

comparing the UMAT and LAM3DNLP codes ..................................... 59

Figure 3.17 Stress vs. strain curves in the Z-direction for [±35°]s IM7/8551-7

comparing the UMAT and LAM3DNLP codes ..................................... 60

Figure 3.18 Stress vs. strain curves in X-direction tension for [±55°]s E-

Glass/MY750 comparing the UMAT and LAM3DNLP codes .............. 61

Figure 3.19 Stress vs. strain curves in X-direction compression for [±55°]s

E-Glass/MY750 comparing the UMAT and LAM3DNLP codes .......... 62

Figure 3.20 Stress vs. strain curves in Y-direction tension for [±55°]s E-

Glass/MY750 comparing the UMAT and LAM3DNLP codes .............. 63

Figure 3.21 Stress vs. strain curves in Y-direction compression for [±55°]s

E-Glass/MY750 comparing the UMAT and LAM3DNLP codes .......... 64

Figure 4.1 Relation of global coordinates to laminate and ply coordinates ............. 69

Figure 4.2 Design methodology flowchart .............................................................. 71

Figure 4.3 Diagram of bending sample.................................................................... 73

Figure 4.4 Plot of safety factor vs. displacement ..................................................... 74

Figure 4.5 Plot of progression of safety factor (SF) ................................................ 75

Figure 4.6 Plot of minimum stiffness ratio (MINCIJ) vs. displacement ................. 76

ix

Figure 4.7 Plot of progression of minimum stiffness ratio (MINCIJ) ..................... 77

Figure 4.8 Diagram of shear sample ........................................................................ 79

Figure 4.9 Contour plot of minimum Cij stiffness ratio, MINCIJ ............................ 81

Figure 4.10 Contour plot of the direction of minimum Cij ratio, MINCIJN .............. 82

Figure 4.11 Plot of CZZ stiffness ratio, CZZR .......................................................... 84

Figure 4.12 Plot of CXY stiffness ratio, CXYR ........................................................ 85

Figure 4.13 Plot of CYY stiffness ratio, CYYR ........................................................ 86

Figure 4.14 Plot of CXX stiffness ratio, CXXR ........................................................ 87

Figure 4.15 Contour plot of number of failures, NOF ............................................... 88

Figure 4.16 Plot of progression of last failed mode, FMODE ................................... 90

Figure 4.17 Plot of progression of last failed ply, FPLY ........................................... 91

Figure 4.18 Diagram of regions ................................................................................. 92

Figure 4.19 Comparison of minimum stiffness ratio MINCIJ [0°/90°]s (left)

vs. [02/±45°]s (right) ............................................................................... 94

Figure 4.20 Plot of the direction of minimum Cij ratio MINCIJN for [02/±45]s

layup ....................................................................................................... 95

Figure 4.21 Plot of CXX stiffness ratio CXXR comparing [0°/90°]s (left) to

[02/±45]s (right) ...................................................................................... 96

Figure 4.22 Plot of CYY stiffness ratio (CYYR) comparing [0°/90°]s (left) to

[02/±45]s (right) ...................................................................................... 97

Figure 4.23 Plot of CZZ stiffness ratio (CZZR) comparing [0°/90°]s (left) to

[02/±45]s (right) ...................................................................................... 98

Figure 4.24 Plot of CXY stiffness ratio (CXYR) comparing [0°/90°]s (left) to

[02/±45]s (right) ...................................................................................... 99

Figure 4.25 Comparison of minimum stiffness ratio MINCIJ, [0°/90°]s (left)

vs. [02/±25°]s (right) ............................................................................. 102

x

Figure 4.26 Plot of the direction of minimum Cij ratio MINCIJN for [02/±25]s

layup ..................................................................................................... 103

Figure 4.27 Plot of CXX stiffness ratio (CXXR) comparing [0°/90°]s (left) to

[02/±25]s (right) .................................................................................... 104

Figure 4.28 Plot of CYY stiffness ratio (CYYR) comparing [0°/90°]s (left) to

[02/±25]s (right) .................................................................................... 105

Figure 4.29 Plot of CZZ stiffness ratio (CZZR) comparing [0°/90°]s (left) to

[02/±25]s (right) .................................................................................... 106

Figure 4.30 Plot of CXY stiffness ratio (CXYR) comparing [0°/90°]s (left) to

[02/±25]s (right) .................................................................................... 107

Figure 4.31 Diagram of open hole sample ............................................................... 109

Figure 4.32 Contour plot of minimum Cij stiffness ratio MINCIJ ........................... 111

Figure 4.33 Contour plot of the direction of minimum Cij ratio MINCIJN ............. 112

Figure 4.34 Contour plot of number of failures, NOF ............................................. 113

Figure 4.35 Plot of CZZ stiffness ratio (CZZR) ........................................................ 114

Figure 4.36 Plot of CXX stiffness ratio (CXXR) ...................................................... 116

Figure 4.37 Plot of CYY stiffness ratio (CYYR) ...................................................... 117

Figure 4.38 Plot of CXY stiffness ratio (CXYR) ...................................................... 118

Figure 4.39 Plot of progression of last failed mode FMODE .................................. 120

Figure 4.40 Plot of progression of last failed ply FPLY .......................................... 121

Figure 4.41 Diagram of regions ............................................................................... 122

Figure 4.42 Plot of CYY stiffness ratio (CYYR) comparing [0°/90°/±45°]s

(left) to [0°/90°/±30°]s (right) ............................................................... 125

Figure 4.43 Plot of last failed ply (FPLY) comparing [0°/90°/±45°]s (left) to

[0°/90°/±30°]s (right) ............................................................................ 126

Figure 4.44 Comparison of minimum stiffness ratio (MINCIJ), [0°/90°/±45°]s

(left) to [0°/90°/±20°]s (right) ............................................................... 127

xi

Figure 4.45 Plot of the direction of minimum Cij ratio MINCIJN for

[0°/90°/±20°]s layup ............................................................................. 128

Figure 4.46 Plot of CXX stiffness ratio (CXXR) comparing [0°/90°/±45°]s

(left) to [0°/90°/±20°]s (right) ............................................................... 129

Figure 4.47 Plot of CYY stiffness ratio (CYYR) comparing [0°/90°/±45°]s

(left) to [0°/90°/±20°]s (right) ............................................................... 130

Figure 4.48 Plot of CZZ stiffness ratio (CZZR) comparing [0°/90°/±45°]s (left)

to [0°/90°/±20°]s (right) ........................................................................ 131

Figure 4.49 Plot of CXY stiffness ratio (CXYR) comparing [0°/90°/±45°]s

(left) to [0°/90°/±20°]s (right) ............................................................... 132

xii

ABSTRACT

Material nonlinearity and progressive ply failure are important

considerations in the finite element modeling of thick section composite structures.

Ply-level anisotropic nonlinearity and ply-based failure criteria require ply-level

stresses and strains, but discretely modeling individual plies in a large scale, thick

section composite structure with hundreds of plies is not computationally feasible. A

method was developed [1-4] to model the material nonlinearity and progressive ply

failure of composite laminates using finite element software without explicitly

simulating individual plies. In this method, the multiple plies of a laminate are treated

as a homogenous, or smeared, material with equivalent material properties. Ply-level

material nonlinearity and progressive failure analysis are incorporated by

dehomogenizing, or unsmearing, the laminate. The LAMPATNL program integrated

this method into a finite element environment [4]. Improvements to this program are

documented and discussed in this work. Also, the LAMPATNL user material is

validated against both the linear and nonlinear material point models.

In this work, a standardized process for designing composite structures

with the LAMPATNL user material is developed. The design methodology uses

newly formulated output parameters, stiffness ratios, to analyze the nonlinear response

and progressive failure of the composite structure. These new parameters greatly

improve the visualization of critical design information of the structure. An example

comparing the original outputs of LAMPATNL to the new outputs is provided to

illustrate the contributions of the stiffness ratio parameters. Two case studies, an open-

xiii

hole sample under multi-axial loading and a compressive shear sample, are evaluated

using the design methodology. Changes to the layups of the laminates in these cases

are made using the insights gained from the design methodology. Coupling

LAMPATNL with a design methodology and new stiffness ratio parameters

demonstrates the utility of progressive failure and nonlinear analysis when applied to

composite structures. The straightforward visualization of critical design information

creates a unique approach to analyzing the design of thick section composites. This

methodology represents a unique contribution to the modeling of composite structures

that is not matched by any current composite model.

1

Chapter 1

INTRODUCTION

As the use of composite materials expands, the demand for an accurate

model of the response and failure of these materials has increased. The adoption of

the finite element (FE) method as a tool to analyze the response of complex structures

under load has greatly improved the understanding and design of these structures.

Linear, elastic materials that are used in many applications are easily handled by any

well developed FE software, but the unique nature of laminated composite materials

have not been fully addressed in commercially available software. Composite

materials are unique due to their ply-level, anisotropic, nonlinear stress-strain

response. It is possible through the load-displacement history of a composite structure

for a ply or plies to fail while the overall structure does not fail. Accounting for first-

ply failure and any subsequent failures, know as progressive failure analysis, is

therefore important in accurately simulating the response of composites. The

incorporation of progressive failure for composite materials in commercial software

has been limited [5]. In this chapter, a review of the state-of-the-art composite

modeling features offered by the most widely available finite element software and

specialized composite-oriented supplemental software is presented. A literature

review on topics related to the progressive damage and failure analysis of composite

structures is also discussed.

2

1.1 Problem statement

Large scale, thick section composite structures cannot be modeled

efficiently using a discrete ply-by-ply approach. For structures with hundreds of

layers, the number of elements through the thickness would be too large to be

computationally efficient. Creating the model and interpreting the ply-by-ply stresses

and strains would be very time consuming and prone to errors. To address the issues

of ply-by-ply analysis, a multi-scale approach is taken to account for the ply-level

response and failure of the composite structure while not explicitly modeling the plies.

To achieve this, multilayered laminates are treated as a homogenous, or smeared,

material with equivalent material properties. Ply-level material nonlinearity and

progressive failure analysis using ply-based failure criteria are incorporated into the

multi-scale approach by dehomogenizing, or unsmearing, the laminate.

An investigation of three commercially available finite element programs,

state-of-the-art composite modeling software products, and recent publications is

presented in this chapter. All these works contain either anisotropic material

nonlinearity, or progressive failure analysis, or the multi-scale “smearing-unsmearing”

approach, but it is shown that none contain all three. The combination of these three

aspects is what makes the modeling approach presented in this work unique. The

objective of this research is to develop a methodology that uses this unique multi-scale

approach in conjunction with finite element analysis to design and analyze thick-

section composite structures.

3

1.2 Benchmarking

1.2.1 State-of-the-art commercial composite models

The most advanced composite modeling capabilities of current,

commercially available, FE software are investigated in this section. ABAQUS,

ANSYS, and MSC.Nastran are three major finite element programs that have

composite modeling capabilities. All three programs have specialized features that

enable the user to construct and analyze composite materials. For the purpose of this

research, the features investigated in these programs include: the ability to have ply-

level nonlinearity in the principle and shear directions, progressive ply failure

capabilities, support for a wide variety of failure criteria, the ability to efficiently

model the layup by homogenizing properties to get an equivalent response, and an

effective way of visualizing this data.

In recent software updates, ABAQUS has incorporated the ability to

model composite layups through the use of conventional or continuum shell layup

sections for shell elements and solid layup sections for solid elements [6]. These

sections are able to analyze composite materials by modeling each ply discretely. The

layup can be comprised of isotropic nonlinear materials, but nonlinear anisotropic

materials are not supported [7]. There is support for progressive failure and damage

evolution laws in ABAQUS, but it is restricted to linear, elastic materials, which can

be anisotropic [8]. Furthermore, progressive failure behavior is restricted to shell

elements and the only damage initiation criterion that can be used is the Hashin criteria

[8]. ABAQUS is able to visualize the damage initiation criteria (tensile and

compressive damage index for both fiber and matrix) and has the ability to label these

4

values by ply. Although limited in scope and restricted to specific element types,

ABAQUS provides a few important modeling features for composite materials.

ANSYS uses both shell (SHELL99, 91, and 181) and solid (SOLID186,

46, 191, and 95) elements to model composite layups [9]. Each shell and solid

element listed has distinct advantages and disadvantages, as well as specific

applications in which they are used. Several of these elements are able to handle

nonlinear material properties [9], with extensive options for different types of

anisotropic nonlinear behavior. The three basic failure criteria that are supported by

default are maximum stress, maximum strain, and Tsai-Wu [10], with the option of

expanding the criteria through user written code. These failure criteria are used to

determine whether the material would have failed and are not coupled to the overall

structural response. ANSYS does contain capabilities for delamination and crack

growth and propagation, but progressive failure for composites in ANSYS is not

possible. Similar to ABAQUS, ply based results are available for viewing in ANSYS,

but there is limited support for critical mode and ply display options.

The programs developed under MSC Software, including Patran, Nastran,

Marc, and several others, have recently begun to add progressive failure capabilities to

their existing composite modeling abilities. In Nastran, a total of seven failure criteria,

including Puck and Hashin, can be used in conjunction with progressive failure [11].

A variety of elements are available for modeling composites, including 2D and 3D

layered shell, solid, and solid shell elements. Material options can be used to create

anisotropic, nonlinear material response. Like other FE software, there are few

options for visualizing and analyzing composite-specific design data that is produced

from progressive failure analysis. The majority of MSC Software’s progressive failure

5

features are included in a separate product, developed in part by NASA, called

GENOA. This product and other notable composite design oriented products are

discussed in the next section.

1.2.2 State-of-the-art for other software products

Before the most recent releases of major FEA programs, there was little

support and options for modeling composite materials. A demand for the accurate

characterization of composite structures spurred the development of several products

that went beyond the available capabilities of these programs. Developed either as an

add-on to software such as ABAQUS, ANSYS, Nastran, and other programs or as a

standalone pre- or post-processing product, features such as progressive damage,

nonlinear material properties, and advanced composite design visualization were able

to be incorporated into the FEA design cycle. While the built-in capabilities of current

FEA programs are catching up, these products still offer some of the most advanced

composite modeling features available.

As mentioned before, GENOA is standalone pre- and post-processing

software that is compatible with Nastran, MSC MARC, LS-DYNA, ANSYS, and

ABAQUS [12]. Offering a multitude of features, GENOA includes multi-scale

progressive failure analysis, the ability to handle nonlinear stress-strain response, and

advanced visualization options. In GENOA, the plies that make up the composite

structures are modeled and displayed separately. In the simulation of structures that

are hundreds of layers thick, the homogenization of composite laminates can improve

the computational efficiency of the model. A homogenizing feature is something that

is missing from this software.

6

Another product that is specifically tailored to simulate composite

materials is Helius:MCT created by Firehole Technologies. Helius was developed as a

user material subroutine for ABAQUS and ANSYS [13]. Through an ABAQUS plug-

in and graphical interface, a composite material that is selected from a database is

turned into a user material (UMAT). This UMAT allows for coupled, progressive

failure of the composite part based on a multi-continuum theory. Nonlinear material

response is limited to the in-plane and out-of-plane shear stiffness [14]. Much like

modeling a composite in ABAQUS, a layered composite section is created using the

UMAT as the constitutive material. The main drawback to this approach is that each

ply is modeled separately, decreasing the computational efficiency when compared to

a homogenized, equivalent composite material. In Helius, the progressive failure

analysis provides visualization of matrix and fiber failure of the structure, but does not

provide indication of whether it is tensile or compressive and there is no shear failure

support. Directionally dependent material nonlinearity and a homogenization

approach are aspects of composites modeling that are not currently available in

Helius:MCT.

There are several additional standalone and integrated products that have

been developed to tackle a range of composite modeling problems. ESAComp,

offering integration with ABAQUS, ANSYS, LS-DYNA, Nastran, NISA, and I-

DEAS, is capable of evaluating several different failure theories of nonlinear

composites and provides informative plots of analysis results through a graphical

interface [15]. NEi Nastran [16] has unique composite analysis features such as the

visualization of critical mode and ply data, progressive ply failure analysis, support for

the latest failure criteria, and nonlinear material models. NISA/Composite [17] is able

7

to display specialized composite analysis parameters and model nonlinear materials

using several available 3D layered elements. The main drawback of all these

programs is the reliance on individual ply modeling and subsequent lack of support for

the homogenizing of properties, which improves the computational efficiency of

modeling of large-scale, thick section, composite structures.

1.2.3 Literature review

A review of recent publications relating to progressive damage and failure

modeling, as well as nonlinear composite material modeling, was conducted to

determine the current state of research in these areas. Many papers deal with both of

these areas and provide innovative ways to utilize them. The search was focused on

trying to find other research that contained a part or all of the key features that have

been determined to effectively model large laminated structures. The key aspects of

the proposed modeling solution are nonlinear material responses that are directionally

dependent, progressive ply failure, and the ability to efficiently model the composite

through homogenized, effective properties.

Numerous researchers have conducted progressive failure analyses using

linear, orthotropic material properties. Using a newly developed damage evolution

criteria, Lapczyk and Hurtado [18] use the progressive damage features within

ABAQUS to model open-hole tension tests of aluminum/composite sandwich

structures. Zhang et al. [19] create a finite element model of grid stiffened plates

subjected to damage based on a modified Hashin criterion. Progressive damage and

delamination are investigated on open-hole samples subjected to tension, compression,

and interlaminar shear. Spottswood and Palazotto [20] use simplified large

displacement and rotation theory to look at the progressive failure of curved composite

8

shells up to and through their snapping point. The effect of two material degradation

models on the response of composite bolted joints is investigated by Huhne et al [21].

Both a constant and continuous degradation approach are incorporated into an

ABAQUS subroutine and compared to experimental data. Xie and Biggers [22] look

at the effect width-to-hole-diameter ratio on open-hole tensile response by using a

progressive damage model that discounts plies instantaneously after failure. This

behavior is programmed into an ABAQUS user material and the ability to tailor the

composite using the results is demonstrated. Buckling and progressive damage of an

open-hole compression sample is simulated by Labeas et al [23]. Using ANSYS, two

different failure theories and two degradation models are compared with experimental

results to determine the best combination. Tserpes et al. [24, 25] created a user

material for ANSYS that uses the Hashin criteria and the ply discount method to

model progressive failure. Variations on this approach were used to parametrically

study laminate-bolt interactions to close agreement.

Nonlinear stress-strain response, especially in the transverse and through-

thickness directions, is an important aspect of composites. There are a number of

papers that incorporate progressive failure with nonlinear response. Schuecker and

Pettermann [26] create a continuum damage model that they compare to results from

the World Wide Failure Exercise (WWFE). Hochard et al. [27] develop an ABAQUS

user material using shear damage and inelastic strain evolution laws. Both static and

fatigue loads are applied to open-hole samples and strain fields are compared to data

obtained from digital image correlation. The progression of failure and post-buckling

response of tapered composites, especially those with ply drop-offs, is studied by

Ganesan and Liu [28]. Using nonlinear ply properties and the maximum stress failure

9

criteria, multiple layups are evaluated against one another. Kilic and Haj-Ali [29]

developed an ABAQUS user material that utilizes Ramberg-Osgood based nonlinear

behavior and Tsai-Wu failure criteria to model variations of both single bolt and open-

hole tension tests. Open-hole and double notched compression samples are modeled

by Basu et al. [30] using layered shell element and a user material in ABAQUS.

Nonlinear material properties are simulated using Schapery theory and fiber rotation

under axial compression. Antoniou et al. [31] simulated thick cylindrical composite

shells under compressive, tensile, and torsion loading in ANSYS. A user material that

had linear, elastic fiber direction response and nonlinear transverse and shear response

was used to create failure envelopes of first and last ply failure for comparisons with

experimental data. Huang [32] developed an ABAQUS user shell section that uses the

Bridging model to degrade the nonlinear fiber and matrix properties upon failure. The

effect of mesh size, element type, and load application are investigated and compared

to experimental results. A unique aspect of this failure analysis is that if a ply

associated with an element fails, that ply is discounted throughout the entire structure.

While there are several examples of both linear and nonlinear behavior

coupled with progressive failure analysis, none of these papers have the ability to

simulate effective properties by homogenizing the laminate into one material. An

important part of modeling is the computational efficiency of the code, so the designer

can quickly analyze many design iterations. This aspect of progressive failure and

composites simulation has not been addressed in previous work and is important when

creating an effective design tool.

10

1.2.4 Selection of FEA software

An investigation of the phrases composite, nonlinear, nonlinear composite,

composite progressive failure, and nonlinear composite progressive failure was

conducted using publications from January 2000 to January 2010. The most widely

mentioned codes in publications containing the key phrases were ABAQUS, ANSYS,

and, to a lesser extent, MSC.Nastran. Table 1.1 shows the percentage of papers with

the phrases that also mention ABAQUS, ANSYS, or Nastran. Throughout all of these

papers, the most widely used program is ABAQUS, ranging from 1% of all papers

with the keyword composite to 17.6% of all papers with the keywords nonlinear

composite progressive failure.

Table 1.1 Percentage of keyword papers containing FEA software

Keywords Abaqus ANSYS Nastran

Composite 1.1% 0.8% 0.2%

Nonlinear 1.3% 0.8% 0.1%

Nonlinear Composite 4.7% 3.0% 0.5%

Composite Progressive Failure 5.6% 2.5% 0.4%

Nonlinear Composite Progressive Failure 17.6% 6.8% 1.3%

Due to its wide use in modeling the progressive damage and failure of

nonlinear composite materials, ABAQUS was selected as the software in which the

homogenization approach and progressive ply failure technique will be implemented.

11

Chapter 2

SOLID MECHANICS APPROACH

The solid mechanics based “LAM” codes were developed for the design

and failure analysis of thick section composite structures. Both linear and nonlinear,

these codes use a multi-scale “smearing-unsmearing” approach and progressive ply

failure to analyze multilayered composite laminates. A background on the

development of these programs and discussion on the key assumptions and procedures,

particularly for the nonlinear codes, are given in this chapter.

2.1 Smearing-unsmearing approach

The analysis of composite structures using finite element techniques is

complicated by the nonlinear, anisotropic ply-level response of these materials.

Furthermore, the failure analysis of a ply or lamina depends upon the stress and strain

state within individual plies. It is therefore important to track the individual ply

stresses and strains in order to accurately model the nonlinear material response and to

be able to apply failure predictions.

One way to accurately calculate the ply-level stresses and strains would be

to model each ply discretely. Using several elements through the thickness of the ply,

a mesh containing all the plies of the structure could be created. While this approach

would most accurately capture the ply-level response, there would be several

disadvantages. A model that is both large in scale and in number of plies would prove

very time consuming to create, run, and analyze. In addition, the vast amount of data

12

produced by a large, complex model would be difficult to interpret. There would also

be a greater chance for errors when creating the model when compared to a

homogenized structure.

A way to address the computational and logistical disadvantages of

discretely modeling each ply is by homogenizing the laminate. Complications arise

when the application of ply-based failure criteria are considered for a homogeneous

material. Ply-level stresses and strains must be extracted from structural stress and

strain profile in order to conduct failure analysis. To deal with these issues, Bogetti et

al. implemented the well established “smearing/unsmearing” approach [1-3].

Equivalent laminate properties are generated by “smearing” the ply-level nonlinear,

anisotropic material behavior of the layup and ply-level stress state needed for failure

analysis is generated by “unsmearing” the elemental stresses and strains.

2.2 Background of the “LAM” codes

The programs LAM3D, LAM3DNLP, LAMPAT, and LAMPATNL are

the four codes in the “LAM” series. The programs use the “smearing-unsmearing”

approach and progressive ply failure analysis to model multilayered composite

laminates on two scales. LAM3D and its nonlinear version LAM3DNLP are codes

that use analytical models to simulate the material point response of laminates.

LAMPAT and LAMPATNL scale up the LAM3D and LAM3DNLP approaches for

use in finite element analysis to evaluate composite structures. A discussion of each

code is provided in this section.

LAM3D was created by Bogetti et al. [1] and uses orthotropic, linear ply

properties and progressive ply failure to simulate the material point response of a

laminate. The code determines the three dimensional effective properties, ultimate

13

strength predictions under mechanical and thermal loading, and effective stress-strain

response due to progressive ply failure under mechanical loading of thick section

composites. Using a specified mechanical load, LAM3D simulates first ply to last ply

failure, calculates effective laminate properties and ply-level stresses and strains after

each failure, and determines the safety factor, critical mode, and critical ply of the

laminate.

The effective property and strength predictions of the LAM3D code for

many composite laminates were validated against theoretical predictions and

experimental results. While LAM3D provided valuable insight into laminate response

due to progressive failure, the code did not support nonlinear ply properties.

Nonlinear material response is more accurate than linear for the transverse and shear

directions of composite materials. In order to address this issue, Bogetti et al. [2]

created the nonlinear, progressive failure code LAM3DNLP. In this code, nonlinear

constitutive relationships for all six principle directions were modeled using Ramberg-

Osgood parameters. Similar to LAM3D, the “smearing-unsmearing” approach and

progressive ply failure methodology are used by LAM3DNLP to simulate the material

point response of a laminate. Both LAM3D and LAM3DNLP were independent

analytical programs developed using FORTRAN and were not coupled to finite

element analysis. The desire to use these material point models in the design of large

composite structures led to the development of LAMPAT and LAMPATNL.

The LAM3DNLP analytical code was ranked third out of 19 participants

in the first World Wide Failure Exercise (WWFE-I) [33]. This exercise was an

assessment of the state-of-the-art of composite failure predictions. LAM3DNLP was

used to predict progressive ply failure in composite laminates for 14 loading cases.

14

These failure predictions were compared to experimental results and rated against the

failure predictions of the other participants [33-35]. The 14 loading cases provided an

experimental validation of the LAM3DNLP failure predictions. The first exercise

focused on failure response of laminates subjected to in-plane loading. The second

exercise (WWFE-II) is currently in progress [36]. This exercise focuses on out-of-

plane failure predictions and, similar to the first exercise, will provide additional

validation for LAM3DNLP.

LAMPAT was created by Bogetti et al. [4] as a way of using the

“smearing-unsmearing” approach and progressive failure analysis of LAM3D with

finite element software packages. LAMPAT was designed as a pre- and post-

processor that created effective composite laminate properties and evaluated the failure

of laminates based on element stress and strain. Originally developed to interface with

PATRAN, it was later expanded to accommodate ANSYS, ABAQUS, DYNA3D, and

NIKE3D. As a preprocessor, LAMPAT uses the orthotropic, linear lamina properties

and the architecture of the laminate to create a set of “smeared” material properties

that represent the effective response of the laminate. Once the composite structure

with effective properties is evaluated, the postprocessor takes global stresses and

strains and creates “unsmeared” ply stresses and strains. The stress and strain

allowables of the lamina are then used to analyze failures within the composite to

provide safety factor, critical mode, and critical ply data. This process is shown in

Figure 2.1.

15

Figure 2.1 LAMPAT process methodology

Implementing LAM3D into finite element software using LAMPAT

demonstrated the power of this approach when designing large scale, thick section

composite structures. To more accurately capture the response of composite materials,

the material nonlinearity features of LAM3DNLP were needed. In LAMPAT, failure

analysis occurred after the stress and strain profile of the structure was complete. This

did not allow for the redistribution of load through the structure after ply failure had

occurred. In order to address this issue, Powers et al. [4] developed the nonlinear,

progressive failure code LAMPATNL. While LAMPAT was developed as a pre- and

post-processing tool, the nonlinear material behavior and progressive failure analysis

of LAMPATNL required closer integration to the finite element package. This was

accomplished by creating a coupled analysis using a user-defined material in

16

ABAQUS. Limited validation and demonstration of the LAMPATNL code was

provided by Powers in [4]. A full validation and development of a design

methodology using LAMPATNL is documented in Chapters 3 and 4.

2.3 Laminated media methodology

The three dimensional laminated media model that is the basis of all the

“LAM” codes was created from an analytical model developed by Chou et al. [37].

The model is used to determine the homogenized, or “smeared”, engineering constants

of a multilayered laminate (see Figure 2.2). Significant features of the theory are

discussed by Bogetti et al. in [1-3] and an overview is given in this section.

Figure 2.2 Laminate configuration [1]

17

Equation 2.1 represents the effective stress/strain constitutive relationship

for the laminate:

*** ijC 2.1

where T*

6

*

5

*

4

*

3

*

2

*

1

* 2.2

T*

6

*

5

*

4

*

3

*

2

*

1

* 2.3

*

66

*

65

*

64

*

63

*

62

*

61

*

56

*

55

*

54

*

53

*

52

*

51

*

46

*

45

*

44

*

43

*

42

*

41

*

36

*

35

*

34

*

33

*

32

*

31

*

26

*

25

*

24

*

23

*

22

*

21

*

16

*

15

*

14

*

13

*

12

*

11

*

CCCCCC

CCCCCC

CCCCCC

CCCCCC

CCCCCC

CCCCCC

Cij 2.4

The barred notation is used to identify that it is in the coordinate system of

the laminate and the star (*) signifies that it is an average value. The relationship

between the global coordinates (X, Y, Z), laminate coordinates ( X ,Y , Z ), and ply

coordinates (1, 2, 3) is shown in Figure 2.3.

18

Figure 2.3 Relation of global coordinates to laminate and ply coordinates

Chou et al. [37] derived a homogeneous representation for the laminate.

The coefficients of the laminate stiffness matrix, *

ijC , are given in Equations 2.5 to 2.7.

Cij* V

kCijkC13

kC3j

k

C33

k

Ci3k V C3 j

k

C33

k1

N

C33

k V Cijk

C33

k1

N

k 1

N

for (i, j = 1, 2, 3, 6) 2.5

0* ijC and 0* jiC for (i=1, 2, 3, 6; j = 4, 5) 2.6

Cij*

V k

kCijk

k1

N

VkV

kC44

k C55

k C45

k C54

k 1

N

k 1

N

for (i, j = 4,5) 2.7

Where the k term refers to the kth

ply of the laminate, Vk is the ratio of the original

thickness of the kth

ply over the original thickness of the entire laminate, and:

k C44

k C45

k

C54

k C55

k C44

k C55

kC45

k C54

k 2.8

19

The assumption is made that the applied mechanical loading acting on the

laminate * is known, not varying through the thickness, and represents the

"average" or "effective" stress. The associated "effective" or "smeared" laminate

strains * can be obtained directly from the inversion of Equation 2.1.

In determining the individual ply-level stresses and strains, two major

assumptions are used. These assumptions are that the in-plane ply strains are equal to

the effective strains of the laminate and that the out-of-plane ply stresses are uniform

through the thickness and equal to the effective stresses in the laminate. The

assumptions are shown in Equations 2.9 and 2.10, respectively.

*

i

k

i for (i = 1, 2, 6 and k = 1, 2, …, N) 2.9

*

i

k

i for (i = 3, 4, 5 and k = 1, 2, …, N) 2.10

The expression derived by Sun and Liao [38] is used to determine the

remaining strain components and is given in Equation 2.11.

3

k

4

k

5

k

C33

k C34

k C35

k

C43

kC44

kC45

k

C53

k C54

k C55

k

1

3

k

4

k

5

k

C31

k C32

k C36

k

C41

kC42

kC46

k

C51

k C52

k C56

k

1

k

2

k

6

k

for (k = 1, 2, …, N) 2.11

The remaining stress components are easily calculated using the relation in

Equation 2.12.

20

1

k

2

k

6

k

C11

k C12

k C13

k C14

k C15

k C16

k

C21

kC22

kC23

kC24

kC25

kC26

k

C61

k C62

k C63

k C64

k C65

k C66

k

1

k

2

k

3

k

4

k

5

k

6

k

for (k = 1, 2, …, N) 2.12

2.4 Nonlinear constitutive relationships

Ply-level material nonlinearity is represented by using the Ramberg-

Osgood equation. Three dimensional nonlinearity is accommodated in all principal

material directions. The longitudinal, transverse, through thickness, out-of-plane

shear, and in-plane shear directions can all have distinct nonlinear behavior. The

Ramberg-Osgood equation, shown below, defines stress as a function of strain and

three parameters.

nn

E

E1

0

0

0

1

2.13

where E0 is the initial modulus, σ0 is the stress asymptote, and n is the shape factor of

the nonlinear stress-strain curve. The effect that these parameters have on the stress-

strain response is shown in Figure 2.4.

21

Figure 2.4 Nonlinear stress-strain response

For computational considerations, it is desired to define the instantaneous

or tangent lamina stiffness as a continuous function of strain. Taking the derivative of

Equation 2.13 with respect to strain, the following expression is obtained:

Et d

d

Eo

1Eo

o

n

11

n

2.14

where Et is the instantaneous or tangent lamina stiffness modulus expressed explicitly

in terms of strain and the three Ramberg-Osgood parameters.

22

2.5 Incremental approach to predicting nonlinear laminate response

The analytical code LAM3DNLP models the nonlinear laminate response

as a set of piecewise linear increments. It predicts laminate stress-strain response

using a specified load increment up until failure. The incremental approach starts with

the initial stiffness as determined by the smeared properties of the laminate. At the

end of each load increment, the stress state from the laminate is unsmeared for ply-

level stresses and strains, failure analysis occurs, the nonlinear constitutive

relationships are updated to determine the stiffness in each principal ply direction, and

effective properties used for the application of the next load increment are obtained by

smearing the laminate. Figure 2.5 provides a representation of the incremental loading

strategy for an arbitrary laminate.

Figure 2.5 Incremental laminate loading methodology [2]

23

Assume that at point (a), corresponding to the end of the nth

stress

increment, the strain and stress state of the laminate is known ( n

i

n

j

** , ). From this

point, the objective is to determine the strain and stress state at point (b) or

( 1*1* , n

i

n

j ). The effective laminate stiffness matrix at the end of stress increment

n, n

ijC * , is computed from an incremental form of the laminated media model

constitutive relation, Equation 2.1. With the increment in load defined, n

i

* , the

corresponding increment in laminate strain, n

j

* , is calculated from an inverse form

of Equation 2.1:

n

i

n

ij

n

j C *1**

2.15

Individual ply stress and strain increments are calculated according to the

equations presented previously. A cumulative summation is maintained to track the

total stress-and-strain levels in each ply of the laminate. The tangent modulus values

for each ply and material direction are calculated according to Equation 2.14 and used

in the determination of the laminate stiffness matrix for the next laminate stress

increment calculation. The entire nonlinear response for the laminate is obtained by

the cumulative sum of all stress and strain increments throughout the entire stress

loading history.

Figure 2.6 illustrates the importance of the incremental approach when

modeling nonlinear material behavior. This figure shows the modeled lamina stress-

strain response with three different load increments applied. The solid line represents

the exact response that is the analytical solution of the Ramberg-Osgood equation.

The larger load increment results in a response that is stiffer than the desired nonlinear

response. As the increment size decreases, the model converges towards the exact

24

response. It is therefore important to select a sufficiently small load increment when

simulating material nonlinearity.

Figure 2.6 Effect of increment size on nonlinear stress-strain response

2.6 Progressive ply failure methodology

Along with ply-level nonlinearity, progressive ply failure will also

influence the effective laminate response to load. Using the stresses and strains of

individual plies, failure predictions and post-failure behavior are modeled for the

25

laminate. Failure criteria that have been incorporated into the “LAM” codes include

Maximum Strain, Maximum Stress, Hydrostatic Pressure Adjusted Maximum Stress,

Tsai-Wu, Christensen, Feng, Hashin, and Von Mises.

The maximum strain failure criterion is commonly used for analysis using

the “LAM” codes [35, 4]. This criterion compares the current strains of each ply (ε1,

ε2, ε3, ε4, ε5, ε6) to the maximum strain allowables (Y1T, Y1C, Y2T, Y2C, Y3T, Y3C,

Y23, Y13, Y12) in each principal direction and assumes a ply fails if these strain

allowables are exceeded. The relations of the strains to allowables are documented in

Equations 2.16 through 2.24.

If ε1>0 and if ε1>Y1T, then the failure mode is fiber tension 2.16

If ε1<0 and if |ε1|>Y1C, then the failure mode is fiber compression 2.17

If ε2>0 and if ε2>Y2T, then the failure mode is matrix tension 2.18

If ε2<0 and if |ε2|>Y2C, then the failure mode is matrix compression 2.19

If ε3>0 and if ε3>Y3T, then the failure mode is matrix tension 2.20

If ε3<0 and if |ε3|>Y3C, then the failure mode is matrix compression 2.21

If |ε4|>Y23, then the failure mode is interlaminar shear 2.22

If |ε5|>Y13, then the failure mode is interlaminar shear 2.23

If |ε6|>Y12, then the failure mode is in-plane shear 2.24

The ply discount method is used by the “LAM” codes to reduce the

material properties of failed plies. In this method, the material stiffness in the

direction of failure is discounted and the Poisson’s ratios associated with that direction

are adjusted. For example, if a ply fails in the 2 direction, the elastic modulus of that

ply in the 2 direction is set to 1% of the current value and this direction is decoupled

26

by setting ν12 and ν23 to zero. Decoupling the ply by zeroing the Poisson’s ratios

prevents unrealistic ply strains caused by Poisson’s effect in the failed ply. Table 2.1

lists which properties are modified for the six different modes of failure.

Table 2.1 Failure mode and property modification

Bogetti describes the progressive failure procedure of the analytical model

in [2]. As the laminate is loaded and laminate strains develop, the individual ply

strains are monitored. When ply failure is predicted in any ply, according to the

maximum strain failure criteria, the incremental loading to that point is stopped and

the entire laminate stress vs. strain response is recorded. The modulus associated with

the particular mode of failure in the failed ply is then reduced according to property

modifications in Table 2.1 and the incremental loading strategy is repeated from the

beginning (all stresses and strains are set to zero). The loading procedure is continued

until the next failure in a ply is detected. The corresponding modulus value is again

discounted, the laminate response is recorded, and the procedure is repeated. This

progressive ply failure response is repeated until final failure is determined, which is

assumed when the laminate looses sufficient stiffness such that it cannot carry any load

27

without undergoing an arbitrarily excessive amount of deformation (greater than 10%

strain).

This progressive failure procedure is effective in capturing the nonlinear

stress-strain response of the laminate, but modifications to this approach are needed

for successful implementation into FEA software. These modifications will be

discussed in the next chapter.

28

Chapter 3

IMPLEMENTATION

LAM3DNLP was implemented into finite element software by Powers et

al. with the creation of LAMPATNL [4]. LAMPATNL is a user material subroutine

(UMAT) used to incorporate the “smearing-unsmearing” approach, nonlinear

anisotropy, and progressive failure analysis into ABAQUS. The subroutine UMAT is

used to define the mechanical constitutive behavior of a material and is called at all

material calculation points of the element. LAMPATNL treats each material

calculation point within the finite element model as though it was a laminate in

LAM3DNLP.

Several implications arise when converting the analytical code into a user

material subroutine. The analytical code was designed to simulate a laminate as a

material point under a prescribed load increment. While it is relatively straightforward

to keep track of the nonlinear response and failure history of a single point, mapping

this procedure for thousands of elements and material points can be complicated. In

addition, there are certain constraints imposed by the finite element method to insure

stability. The constraints are based on thermodynamic stability conditions. Changes

to the material response and the key assumptions of the laminated media methodology

are required. Powers [4] briefly describes these implications and changes, but an

investigation into these issues is discussed below.

29

3.1 Theoretical consideration

3.1.1 Ply-level stress and strain calculations

The two major assumptions of the theory used in the “LAM” codes are

stated in Chapter 2. The first assumption is that the in-plane ply strains are equal to

the effective strains of the laminate and is shown in Equation 3.1. The second

assumption is that the out-of-plane stresses are uniform and equal to the effective

stresses in the laminate and is expressed in Equation 3.2. In these equations, the k

term refers to the kth

ply of the laminate.

*

i

k

i for (i = 1, 2, 6 and k = 1, 2, …, N) 3.1

*

i

k

i for (i = 3, 4, 5 and k = 1, 2, …, N) 3.2

All the remaining ply strains (3, 4, and 5) and ply stresses (1, 2, and 6) are

determined by the expressions given by Sun and Liao [38], shown in Equations 3.3

and 3.4, respectively.

3

k

4

k

5

k

C33

k C34

k C35

k

C43

kC44

kC45

k

C53

k C54

k C55

k

1

3

k

4

k

5

k

C31

k C32

k C36

k

C41

kC42

kC46

k

C51

k C52

k C56

k

1

k

2

k

6

k

for (k = 1, 2, …, N) 3.3

1

k

2

k

6

k

C11

k C12

k C13

k C14

k C15

k C16

k

C21

kC22

kC23

kC24

kC25

kC26

k

C61

k C62

k C63

k C64

k C65

k C66

k

1

k

2

k

3

k

4

k

5

k

6

k

for (k = 1, 2, …, N) 3.4

30

The ply stresses and strains are used to calculate the tangential stiffness

properties for each ply. These ply-level properties are “smeared” to determine the

effective stiffness of the laminate for that increment. Ply stresses and strains are also

needed to apply ply-based failure criteria for progressive failure in the structure.

The procedure for calculating the ply-level stresses and strains for the

nonlinear analytical code is outlined in Figure 3.1. In LAM3DNLP, the three

dimensional load increment for the laminate, in the form ( yzxzxyzyx ,,,,, ), is

input into the program. After initializing laminate strain, ply strains, and ply stresses,

the program calculates the ply-level tangential stiffness properties, laminate stiffness

matrix, and ply-level stiffness matrices based upon the current ply strains. The

laminate stiffness matrix and laminate load increment are then used to calculate the

laminate strain increment. Using the assumption in Equation 3.1, the in-plane ply

strain increment is set equal to the in-plane laminate strain increment. Similarly for

the assumption in Equation 3.2, the out-of-plane ply stress increment is set equal to the

out-of-plane laminate stress increment. The out-of-plane ply strain increment is

calculated using the ply-level stiffness matrices, the in-plane laminate strain increment,

and the out-of-plane laminate stress increment using Equation 3.3. The in-plane ply

stress increment is calculated similarly using Equation 3.4. The increments for ply

stress, ply strain, and laminate strain are used to update their respective values and the

procedure is continued for the next load increment. This procedure demonstrates the

direct implementation of the assumptions of Chou’s theory when calculating the ply-

level stress and strain components in LAM3DNLP.

31

Figure 3.1 Ply stress and strain calculation procedure for LAM3DNLP

32

Implementing LAM3DNLP into the LAMPATNL UMAT requires

modifications to the procedure shown in Figure 3.1. In ABAQUS, the user material is

required to output the stiffness matrix and update the stress tensor at the material

calculation point at the end of the increment. These values must be determined from

the current laminate stress, the current laminate strain, and the laminate strain

increment. The procedure for calculating the stiffness matrix and stress tensor using

ply-level stresses and strains in LAMPATNL is outlined in Figure 3.2.

The major difference between the LAMPATNL procedure and the

LAM3DNLP procedure occurs when determining the tangential stiffness properties of

the plies. In LAM3DNLP, the ply strains that are initially zero are recorded and

updated throughout the procedure. These strains are used to calculate ply-level

material properties and ply and laminate stiffness matrices. The LAMPATNL user

material subroutine does not record the ply-level strain data from increment to

increment due to the “smearing” approach in the code. LAMPATNL must calculate

the ply-level material properties and ply and laminate stiffness matrices at the

beginning of the increment without the ply-level strains. To accomplish this, the ply

strains are set equal to the laminate strains as shown in Equation 3.5.

*

i

k

i for (i = 1, 2, 3, 4, 5, 6 and k = 1, 2, …, N) 3.5

This enables LAMPATNL to calculate the ply and laminate stiffness

matrices at the beginning of the increment, update the stress tensor for the increment,

and determine the laminate stiffness matrix at the end of the increment. Ply stresses

used for stress based failure criteria are determined analytically using the ply strains

and Ramberg-Osgood relationship.

33

The out-of-plane ply strains are taken directly from the laminate strains in

LAMPATNL while the out-of-plane strains in LAM3DNLP are calculated using the

stiffness matrix, out-of-plane laminate stresses, and in-plane laminate strains. These

changes do not affect the calculated stress-strain response of laminate made of one

material through the thickness or a laminate subjected to in-plane loading. The

implication of this change for hybrid composites, with different material through the

thickness, is a subject for future work.

34

Figure 3.2 Ply stress and strain calculation procedure for LAMPATNL

35

3.1.2 Poisson’s ratio

The response of the composite laminate is modeled using the Ramberg-

Osgood equation given in Equation 2.13. This stress-strain response is nonlinear and,

when coupled with the incremental approach, becomes a piecewise linear response. At

each load step, the instantaneous tangent moduli for the extensional and shear

directions are calculated using Equation 2.14. Over each load step, the material is

considered to be a linear, elastic material.

The numerical stability for calculations of a linear, elastic, orthotropic

material requires the following five conditions to be met [39]:

0,,,,, 231312321 GGGEEE

3.6

2/1

2

112

E

E

3.7

2/1

3

113

E

E

3.8

2/1

3

223

E

E

3.9

021 133221311332232112

3.10

At the start of the analysis, these conditions are satisfied, but the

incorporation of ply-level progressive failure can cause conditions 3.7, 3.8, and 3.9 to

become unsatisfied. To correct this problem, the Poisson’s ratio for each ply is

continuously updated throughout the simulation. The current moduli are used in

conjunction with the original minor Poisson’s ratios (ν21, ν31, ν32), which are stored at

36

the beginning of the analysis, to calculate the updated Poisson’s ratios and ensure

stability. The reciprocity relationships used for this are:

T

Tic

E

E

3

23223

3.11

T

Tic

E

E

3

13113

3.12

T

Tic

E

E

2

12112

3.13

where c is the current Poisson’s ratio,

i is the initial Poisson’s ratio, and TE is the

tangential, or instantaneous, elastic modulus at the load increment. These adjustments

ensure that the material stability of the UMAT is satisfied through the entire

simulation. The LAMPATNL work of Powers et al. [4] contains these adjustments

but does not specifically document them.

3.2 Procedural considerations

3.2.1 Progressive failure adaption for FEA

The LAM3DNLP approach was developed to analyze the material point

response of a laminate. Modifications to the post failure behavior of the analysis are

made when implementing the code in LAMPATNL. In LAM3DNLP, a specified load

increment is applied to the laminate up until failure. Once failure occurs, the

necessary adjustments to the failed plies, such as elastic modulus reduction and

Poisson’s ratio adjustment, are made and the analysis of the modified material point

restarts from the first load increment. This process is illustrated in the flowchart

shown in Figure 3.3.

37

Figure 3.3 LAM3DNLP progressive failure procedure

The procedure creates multiple stress-strain curves and load “drops”

between them, as shown in of Figure 3.4. This figure shows the stress-strain response

and progressive failure for this quasi-isotropic layup of S2 Glass/Epoxy. First ply

failure is represented in the first iteration. Ultimate or last ply failure is defined as the

failure that occurs at the highest stress state. In this case, the third iteration is

determined to be representative of last ply failure.

38

0

100

200

300

400

500

600

0.00% 0.50% 1.00% 1.50% 2.00% 2.50% 3.00% 3.50%

Iteration 1

Iteration 2

Iteration 3

LAM3DNLP Output

Figure 3.4 Stress-strain curves of [0°/90°/±45°]s S2 Glass/Epoxy laminate

The effective stress-strain response of the laminate is the combination of

the first three iterations. This output is represented as the circled LAM3DNLP stress-

strain response in Figure 3.4. This iterative procedure is effective in capturing the

immediate loss of load carrying capability that is seen in experimental stress-strain

curves, but the process of restarting the analysis from the first increment is not

practical for finite element applications.

Changing this approach when implementing LAMPATNL was done

simply by continuing the analysis with reduced material properties at the current

stress-strain state after failure has occurred. This change is reflected in the

LAMPATNL flowchart of Figure 3.5. Comparing this procedure to LAM3DNLP

39

procedure in Figure 3.3, the stress and strain state of the laminate is no longer

reinitialized and laminate is not restarted from the first increment.

Figure 3.5 LAMPATNL progressive failure procedure

40

The process changes the predicted stress-strain response to that shown as

the UMAT curve of Figure 3.6. Instead of abrupt drops in load, ply failure causes a

reduction in the overall modulus of the material. This can be seen in the slight

deflection of the curve at 0.35% and 1% strain and the noticeable deflection at 3.2%

strain. To facilitate a one-to-one validation of the UMAT and the analytical code,

simple modifications to the underlying FORTRAN code of the original LAM3DNLP

were made to incorporate this procedure.

0

100

200

300

400

500

600

700

0.00% 0.50% 1.00% 1.50% 2.00% 2.50% 3.00% 3.50%

LAM3DNLP

LAMPATNL

Figure 3.6 Stress-strain curves of [0°/90°/±45°]s S2-Epoxy laminate -

LAM3DNLP output and ABAQUS UMAT output

41

3.2.2 Tracking progressive failure

Improvements to the tracking of failure progression were added to the

LAMPATNL user material subroutine developed by Powers et al. In LAM3DNLP,

one material point is simulated, but LAMPATNL must account for upwards of several

thousand elements with multiple integration points. The user material subroutine is

called at every integration point during each load increment within the analysis.

During each call, the UMAT receives the strain state of that integration point and

determines the stiffness matrix of the material in order for the program to calculate the

stress. To accomplish this, the UMAT must know the failure history of the laminate at

that integration point.

The failure history tracks which ply and in which direction failure has

occurred. This requires the failure history to be element, integration point, ply, and

direction specific (Figure 3.7). With a model containing hundreds of plies each with

six moduli (E1, E2, E3, G12, G13, G23) and thousands of elements each with multiple

integration points, tracking all of this necessary information can be difficult and

memory intensive. It is accomplished by saving a large array that holds the failure

history and is sorted by ply number, element number, and integration point number.

42

Figure 3.7 Tracking failure history is mode, ply, integration point, and element

specific

3.2.3 Selection of elements

Selecting the correct element is important when simulating the behavior of

complex structures. The “LAM” codes were developed to analyze thick section

composite structures. To match the analysis capability of these programs, it is

important for LAMPATNL to be able to interface with three-dimensional elements.

LAMPATNL was developed by Powers et al. to support the use of two dimensional,

solid (continuum), axisymmetrical 4-node bilinear (CAX4) elements. These elements

were first supported because of the need for simulating cylinders and other

axisymmetric structures. The user material was also able to interface with elements

with four term stress and strain tensors of the form ( xyzyx ,,, ) and

( xyzyx ,,, ). To improve the three dimensional modeling capabilities of

43

LAMPATNL, support of 8-node linear brick elements (C3D8 in ABAQUS) was

added. This support extends to any element that has six term stress and strain tensors

in the forms ( yzxzxyzyx ,,,,, ) and ( yzxzxyzyx ,,,,, ).

It is important to investigate which of these supported elements are best

for the accurate simulation of thick-section composite structures and which provide

easily interpreted visualizations of key outputs. Modeling non-axisymmetric

structures leads to the use of the three-dimensional 8-node linear brick elements.

There are several options available for this node type, including hybrid with both

linear and constant pressure, the support of incompatible modes, and reduced

integration.

The reduced integration option for this element has several advantages and

disadvantages. Fully integrated C3D8 elements contain 8 integration points per

element and reduced integration C3D8R elements contain only 1 integration point.

Using reduced integration elements would result in one eighth of the number of

calculations for the same amount of elements. The resulting simulation would be less

accurate, especially in cases where reduced integration elements have historically

exhibited problems, such as bending. Elements with reduced integration are also

prone to hourglassing and controls must be used to prevent undesirable distortions.

The drawbacks of reduced integration elements are balanced by less

computational cost and simplified visualizations. Figure 3.8 shows visualization of a

model using fully integrated elements on the left and reduced integration elements on

the right. For this case, the output is direction of the minimum Cij ratio, MINCIJN,

which can be integer values between zero and six. Non-integers, negative numbers

represented as black areas, and number greater than six (grey areas) do not reflect

44

correct values. These incorrect values are a product of displaying extrapolated values,

derived from integration points, at the nodes of full integration elements. For this

output, the values can be discontinuous from one element to the next, yet the nodal

extrapolation assumes these values to be continuous. Using reduced integration

elements avoids problems associated with the visualization of extrapolated nodal

values.

Figure 3.8 Comparison of visualization of full integration (left) and reduced

integration (right) elements

Careful consideration was taken to ensure all the concerns associated with

reduced integration elements are addressed and the implementation of LAMPATNL in

conjunction with these elements was correct. Using a user material with reduced

integration elements required hourglass stiffness controls. To ensure the hourglass

45

stiffness of the model was set to an appropriate value, the elastic strain energy was

compared to the hourglass energy calculated by the finite element program. The

hourglass energy should be less than 10% of the elastic strain energy to ensure no

hourglassing had occurred. The calculation of elastic strain energy was added to

LAMPATNL to make this comparison.

3.3 Validation of LAMPATNL

Validating the LAMPATNL user material subroutine begins with

modeling simple composite layups using linear ply properties and no progressive

failure. These single element models are validated with comparisons to the material

point response calculated by LAM3D. Matching the stress-strain curve produced by

the UMAT using linear ply properties to that created by LAM3D ensures that the user

material “smears” the ply properties of the laminate correctly. While the stress-strain

curve comparison is straightforward, comparing the output parameters created by each

program is more complex. The parameters calculated by LAM3D are the safety factor,

critical mode, and critical ply. LAMPATNL also produces these parameters and 13

additional parameters, but there are major differences in the meaning of the parameters

between the two.

For the linear case, safety factor is defined as the ultimate load that a

laminate could sustain, based on the allowable amount of progressive failure, divided

by the applied loading on the laminate. The safety factor is therefore a scalar multiple

of the applied loading. The process of determining this value starts by calculating the

ratio of the strain allowable to ply strain for each ply and each direction. The lowest

ratio is called the safety factor for that iteration and the ply and direction in which the

lowest ratio occurs is called the critical ply and critical mode respectively. The ply and

46

direction are next flagged for failure and the stiffness in the critical direction of that

ply is discounted. New linear “smeared” properties for the laminate are then

calculated and the process repeats. A new safety factor, critical mode, and critical ply

are calculated and recorded for the next iteration, the plies are discounted, and

“smeared” properties are updated. Iterations continue until the structural integrity of

the laminate has been compromised. At the end of the analysis, the highest safety

factor of all the iterations is determined to be last ply failure and the mode and ply of

that highest safety factor are reported as critical mode and ply for the element.

The process for determining the safety factor in nonlinear, progressive

failure analysis is not as straightforward. First, stress and strains in nonlinear analysis

cannot be linearly scaled to larger values because this analysis uses the incremental

approach for determining stress from strain. Secondly, the progressive nature of the

analysis requires that plies must exceed their strain allowable to be flagged for failure.

In the linear analysis, the lowest ratio of the strain failure allowable to the ply strain is

designated for failure and the ply is discounted. In nonlinear, progressive failure

analysis, the safety factor is defined as the lowest ratio of failure allowable to ply strain

at the current level and the critical ply and mode are the ply and direction of this

lowest ratio. This determines which ply and direction are closest to failure, not which

ply and direction would cause last ply failure. Because of these differences, it is not

appropriate to directly compare values of the variables calculated in LAMPATNL to

those calculated in the LAM3D.

An outline of the examples used to validate the LAMPATNL user material

subroutine is shown in Table 3.1. Using linear ply properties and no progressive

failure, two laminates are analyzed with LAMPATNL and compared to LAM3D

47

analysis. With nonlinear ply properties and progressive failure, LAMPATNL is

validated against LAM3DNLP for several materials and layups. For all of these

examples, laminate stress-strain response is compared between the UMAT and

analytical models. Because of the differences in safety factor calculations, this value

cannot be directly compared.

Table 3.1 Examples provided for LAMPATNL validation

Layup Material Load (MPa) Load Direction Linear/Nonlinear Progressive Failure

[0/90/±45] S-Glass Epoxy 100 X Tension Linear No

[±30] T300/PR-319 150 Y Tension Linear No

[0] S-Glass Epoxy 1700 1 Tension Nonlinear Yes

[0] S-Glass Epoxy 65 2 Tension Nonlinear Yes

[0] S-Glass Epoxy 50 3 Tension Nonlinear Yes

[0/90/±45] S-Glass Epoxy 800 X Tension Nonlinear Yes

[±35] IM7/8551-7 425 X Tension Nonlinear Yes

[±35] IM7/8551-8 160 Y Tension Nonlinear Yes

[±35] IM7/8551-9 85 Z Tension Nonlinear Yes

[±55] E-Glass/MY750 125 X Tension Nonlinear Yes

[±55] E-Glass/MY750 -200 X Compression Nonlinear Yes

[±55] E-Glass/MY750 350 Y Tension Nonlinear Yes

[±55] E-Glass/MY750 -225 Y Compression Nonlinear Yes

3.3.1 Validation of LAMPATNL for linear materials

In the first example, an S-Glass/Epoxy laminate with a [0°/90°/±45°]s

layup is evaluated in simple tension load of 100 MPa in the X-direction. The linear

ply level properties of this material for LAM3D are shown in Table 3.2. Maximum

strain is the failure criteria that is used for this example and the failure allowables are

also shown in Table 3.2.

48

Table 3.2 Linear ply properties of S-Glass/Epoxy

E1 52000 ε1T

3.270%

E2 19000 ε1C

2.210%

E3 19000 ε2T

0.263%

G12 6700 ε2C

1.500%

G13 6700 ε3T

0.263%

G23 6700 ε3C

1.500%

ν12 0.30 γ12 4.000%

v13 0.30 γ13 4.000%

v23 0.42 γ23 0.590%

Ply Properties Strain Allowables

The ply properties input into the nonlinear user material must be

linearized, which is achieved by setting the stress asymptote and shape factor

Ramberg-Osgood parameters to large values. Table 3.3 contains the LAMPATNL

input values for the linearized material properties. Additionally, to disable the

progressive failure capabilities of the code, maximum strain allowables are set to 10%.

Table 3.3 Ply properties for linear S-Glass/Epoxy

S-Glass/Epoxy

E0 (GPa)

0 (GPa)

n

0.3 0.42— — — 0.3

100 100

10 10 10 10 10 10

100 100 100 100

13 23

52 19 19 6.7 6.7 6.7

1 2 3 12

The linear stress-strain curves of both LAMPATNL and the LAM3D

results are shown in Figure 3.9. The calculated stress-strain responses from these

49

codes are coincident as expected. Because of the reasons stated in the previous

section, the direct comparison of safety factor, critical mode, and critical ply are not

performed.

0

20

40

60

80

100

120

0.00% 0.05% 0.10% 0.15% 0.20% 0.25% 0.30% 0.35% 0.40%

Strain (%)

Str

ess (

MP

a)

LAM3D

LAMPATNL

Figure 3.9 Stress vs. strain curves in the X-direction for [0°/90°/±45°]s S-

Glass/Epoxy comparing the LAM3D and LAMPATNL codes

For the next example, T300/PR-319 in a [±30°]s layup is subjected to a

tensile load of 150 MPa in the Y-direction. The ply properties used for the material

are listed in Table 3.4 below, with the σ0 and n values not used for input into LAM3D.

The resulting linear stress-strain curves are shown in Figure 3.10 and are again

coincident.

50

Table 3.4 Ply properties for linear T300/PR-319

T300/PR319

E0 (GPa)

0 (GPa)

n

10 10

— — — 0.318 0.318 0.5

10 10 10 10

1.33 1.86

100 100 100 100 100 100

129 5.6 5.6 1.33

1 2 3 12 13 23

0

20

40

60

80

100

120

140

160

180

0.00% 0.50% 1.00% 1.50% 2.00% 2.50% 3.00% 3.50% 4.00%

Strain (%)

Str

ess (

MP

a)

LAM3D

LAMPATNL

Figure 3.10 Stress vs. strain curves in the Y-direction for [±30°]s T300/PR-319

comparing the LAM3D and LAMPATNL codes

3.3.2 Validation of LAMPATNL for nonlinear materials

Comparisons between the LAMPATNL user material and LAM3D were

made by suppressing the nonlinear and progressive failure features of the user

material. In order to completely validate its proper function, the user material model

51

must include these features. The next step in validating the UMAT is to match results

using a single element model with those that are output by the analytical code

LAM3DNLP. Changes to the post-failure loading procedure for the analytical code

were detailed in the Procedural considerations sections of this chapter. These changes

facilitate a one-to-one comparison of stress-strain curves, critical modes, and critical

plies between the user material and LAM3DNLP.

The Ramberg-Osgood parameters that are used to model the nonlinear

properties for each material used in the examples are provided in Table 3.5 [2, 36].

The maximum strain failure criterion is used for every example and the strain

allowables are shown in Table 3.6 [2, 36].

52

Table 3.5 Ramberg-Osgood parameters

Material &

Its Parameters

S-Glass/Epoxy

E0 (GPa)

σ0 (GPa)

n

ν

T300/PR319

E0 (GPa)

σ0 (GPa)

n

ν

IM7/8551-7

E0 (GPa)

σ0 (GPa)

n

ν

E-Glass/MY750

E0 (GPa)

σ0 (GPa)

n

ν

E-Glass/LY556

E0 (GPa)

σ0 (GPa)

n

ν

AS4/3501-6

E0 (GPa)

σ0 (GPa)

n

ν 0.28 0.4— — — 0.28

23

6.7

100

10

0.42

for Constitutive Modeling

Spatial Directions

231 2 3 12 13

2.365

1.86

100

10

0.5

23

2.8

100

10

0.278

0.5

23

5.79

100

10

0.4

23

6.32

100

— — — 0.278

0.076

10 10 10 1.85 1.85

100 100 100 0.076

6.8

53.5 17.7 17.7 6.36 6.36

1 2 3 12

126 11 11 6.8

0.278

1 2 3 12 13

— — — 0.278

0.077

10 10 10 1.8 1.8

100 100 100 0.077

6.42

1 2 3 12 13

45.6 16.2 16.2 6.42

2.018

— — — 0.34 0.34

10 3.604 3.604 2.018

0.089

165 9.01 9.01 5.6

100 0.193 0.193 0.089

1 2 3 12

— — — 0.318

100 0.135 0.135 0.108

10 4.728 4.728 4.872

129 5.6 5.6 1.33 1.33

10

0.4

13 23

0.108

4.872

0.318

13

5.6

0.3

10 3.601 3.601 2.365

— — — 0.3

1 2

6.7

100 0.189 0.189 0.071 0.071

52 19

100 100 100 0.097

10 10 10 1.96 1.96 10

3 12 13

3.93

0.097 100

19 6.7

Table 3.6 Maximum strain failure allowables

S-Glass/Epoxy 3.27 -2.21 0.33 -1.5 0.263 -1.5 0.59 4 4

T300/PR319 1.07 -0.74 0.43 -2.8 0.43 -2.8 1.5 8.6 8.6

IM7/8551-7 1.551 -0.96 0.87 -3.2 0.755 -3.2 2.1 5 5

E-glass/MY750 2.81 -1.75 0.25 -1.2 0.25 -1.2 4 4 4

E-glass/LY556 2.13 -1.07 0.2 -0.64 0.2 -0.64 3.8 3.8 3.8

AS4/3501-6 1.38 -1.18 0.44 -2 0.44 -2 2 2 2

Y13 (%) Y12 (%)Y2C (%) Y3T (%) Y3C (%) Y23 (%)Material Y1T (%) Y1C (%) Y2T (%)

53

Using a single element model for each example, the laminate is loaded in

the X-direction until failure has occurred. This process is similarly repeated for the Y-

and Z-directions. The first comparison between the LAMPATNL user material and

LAM3DNLP is a simple unidirectional layup of S-Glass/Epoxy. As with all of the

materials tested, this material is linear in the fiber direction. The tensile failure strains

in the 2- and 3-directions for this material are relatively small, which results in the

nonlinear responses of the material in those directions appearing to be linear. Figure

3.11 shows the stress-strain response of the unidirectional material in the 1 direction.

As expected, the UMAT curve is coincident with the analytical model and both reach

failure at 3.27% strain.

54

0

200

400

600

800

1000

1200

1400

1600

1800

0.00% 0.50% 1.00% 1.50% 2.00% 2.50% 3.00% 3.50%

Strain (%)

Str

ess

(M

Pa

)

LAM3DNLP

UMAT

Figure 3.11 Stress vs. strain curves in the 1-direction for [0°] S-Glass/Epoxy

comparing the UMAT and LAM3DNLP codes

The stress-strain response in the 2-direction is shown in Figure 3.12.

Failure in this direction occurs at 0.33% strain and this is reflected in the results.

Figure 3.13 shows the response of this material in the 3-direction, with failure

occurring at 0.26% strain in both the LAMPATNL and LAM3DNLP.

55

0

10

20

30

40

50

60

70

0.00% 0.05% 0.10% 0.15% 0.20% 0.25% 0.30% 0.35%

Strain (%)

Str

ess (

MP

a)

LAM3DNLP

UMAT

Figure 3.12 Stress vs. strain curves in the 2-direction for [0°] S-Glass/Epoxy

comparing the UMAT and LAM3DNLP codes

56

0

10

20

30

40

50

60

0.00% 0.05% 0.10% 0.15% 0.20% 0.25% 0.30%

Strain (%)

Str

ess (

MP

a)

LAM3DNLP

UMAT

Figure 3.13 Stress vs. strain curves in the 3-direction for [0°] S-Glass/Epoxy

comparing the UMAT and LAM3DNLP codes

The next example is an S-Glass/Epoxy laminate in a [0°/90°/±45°]s, or

quasi-isotropic, layup. In this example, the response to load applied in the X-direction

is the same as response to load applied in the Y-direction, so only one plot is shown

for these two load cases. The nonlinear material response and progressive failure of

the laminate is shown in Figure 3.14. Transverse tensile failure in the 90° plies occurs

in the laminate at 0.35% strain and in the ±45° plies at 1% strain. These failures

produce slight deflections in the stress-strain response. Some additional failure has

little effect on the overall stiffness of the laminate in this direction. Longitudinal

tensile failure in the 0° plies occurred at 3.3% strain that caused a dramatic decrease in

the stiffness of the laminate. The final failure recorded for this case is longitudinal

57

compressive failure in the 90° plies. Both the analytical code and the user material

yield the same results for this test case, continuing to validate the correct function of

the user material.

Figure 3.14 Stress vs. strain curves in the X-direction for [0°/90°/±45°]s S-

Glass/Epoxy comparing the UMAT and LAM3DNLP codes

The out-of-plane stiffness of the laminate is not significantly affected by

its layup and there is no progressive failure in this direction because first ply failure is

catastrophic. The stress-strain response in the Z-direction is the same as that shown in

58

Figure 3.13 because the material used for this laminate is the same used in the first

example.

A laminate comprised of IM7/8551-7 in a [±35°]s layup is used in the next

validation test case. In this composite layup, the nonlinear transverse and shearing

directions are a contributing factor to the response in X-direction loading. The stress-

strain curves in Figure 3.15 do not show abrupt discontinuities or changes in stiffness

outside those normally associated with nonlinear materials, suggesting that the failures

did not significantly affect the response of the laminate in this direction.

Figure 3.15 Stress vs. strain curves in the X-direction for [±35°]s IM7/8551-7

comparing the UMAT and LAM3DNLP codes

59

Subjecting the same laminate to a load in the Y-direction yields a response

that is both highly nonlinear and significantly effected by progressive ply failure. In

Figure 3.16, the stress-strain response deviates from the nonlinear behavior to a more

compliant response at approximately 1.7% strain. This failure corresponds to

transverse tensile failure in the ±35° plies. Ultimate failure in this laminate is in-plane

shearing in all the plies. The results from the user material once again mirror the

results obtained by LAM3DNLP.

Figure 3.16 Stress vs. strain curves in the Y-direction for [±35°]s IM7/8551-7

comparing the UMAT and LAM3DNLP codes

60

The response of this laminate in the Z-direction (Figure 3.17) is linear

until failure. Once again, the small tensile failure strain of the material limits its

response to the linear regime of the nonlinear curve. Both the user material and

LAM3DNLP responses continue to match.

0

10

20

30

40

50

60

70

80

90

0.00% 0.10% 0.20% 0.30% 0.40% 0.50% 0.60% 0.70% 0.80%

Strain (%)

Str

ess

(M

Pa

)

LAM3DNLP

UMAT

Figure 3.17 Stress vs. strain curves in the Z-direction for [±35°]s IM7/8551-7

comparing the UMAT and LAM3DNLP codes

The final example for validation is an E-Glass/MY750 composite in a

[±55°]s layup. This example tests the X- and Y-direction response of the laminate in

both tension and compression. The material properties, layup, and load scenario are

61

the same as those studied in the first World Wide Failure Exercise (WWFE-I) (see test

case 9) [2]. The details of this exercise are discussed in the Background of the “LAM”

codes section of Chapter 2.

The first validation for this laminate is tension in the X-direction. In

Figure 3.18, the laminate has an almost purely linear response up to around 0.5%

strain then a noticeable shift to nonlinear response after. This shift is caused by

transverse tensile failure in all the plies of the laminate. The coincidence of the curves

further validates the results of the UMAT.

Figure 3.18 Stress vs. strain curves in X-direction tension for [±55°]s E-

Glass/MY750 comparing the UMAT and LAM3DNLP codes

62

The second validation is compression in the X-direction, shown in Figure

3.19. Like previous examples, the LAM3DNLP results and the user material results

match. Transverse compressive failure in the plies around 2.5% strain causes the

laminate to become compliant. Ultimate failure in this test case is a result of shear

failure in the plies.

Figure 3.19 Stress vs. strain curves in X-direction compression for [±55°]s

E-Glass/MY750 comparing the UMAT and LAM3DNLP codes

Tension in the Y-direction is next investigated for validation and shown in

Figure 3.20. The response of the laminate is nonlinear with no noticeable stiffness

63

changes that can be attributed to progressive failure. In-plane shear failure in this

laminate around 2.25% strain does not significantly affect the response in this

direction. Ultimate failure in this test case is a result of transverse compressive failure.

The final example in Figure 3.21 is compression in the Y-direction. A

change in stiffness around 1.25% strain is a result of transverse tensile failure in the

plies. This failure has a small effect on the response of the laminate in this direction.

In-plane shear failure in all plies is the ultimate failure in this case. Similar to all other

validation examples, the UMAT stress-strain curves match those produced by the

analytical LAM3DNLP code.

Figure 3.20 Stress vs. strain curves in Y-direction tension for [±55°]s E-

Glass/MY750 comparing the UMAT and LAM3DNLP codes

64

Figure 3.21 Stress vs. strain curves in Y-direction compression for [±55°]s

E-Glass/MY750 comparing the UMAT and LAM3DNLP codes

3.4 Validation summary

The comparison of the LAMPATNL user material output using a single

element model to LAM3D for linear properties provided the first step towards

validating the proper function of the subroutine. Four example cases were shown to

validate the nonlinear and progressive failure aspects of the user material function

correctly by generating the same output as LAM3DNLP. The validated LAMPATNL

user material can now be utilized in the design of composite structures. The procedure

of incorporating the user material into the design process of thick section composites is

documented in the next chapter.

65

Chapter 4

CASE STUDIES

LAM3DNLP and LAMPATNL were developed to analyze symmetric,

thick section composite laminates and structures with ply level material nonlinearity

and failure. In this chapter, two case studies are presented and examined with the

LAMPATNL user material. The first case study is a simple block in compression that

is partially fixed on one edge, inducing in-plane shear stresses. The second case study

is an open hole sample subjected to simultaneous loading in the X- and Y-directions.

The open hole sample is commonly used to check the effectiveness of composite

failure models [18, 19, 21-25, 27, 29, 30]. An example illustrating the improvements

and contributions to the LAMPATNL design method over its original version is also

provided.

4.1 LAMPATNL output parameters

A total of 16 parameters are output by the UMAT, so filtering and

processing this data is an important part of the design process. Several useful output

parameters are calculated for each element in the model in addition to the safety factor,

critical mode, and critical ply parameters produced by the original LAMPATNL user

material [4]. A full list and brief description of each parameter is provided in Table

4.1. These history dependent parameters are calculated and stored throughout the

incremental solution.

66

Table 4.1 List of state dependent parameters

SDV1 CXXR CXX Ratio, X Stiffness

SDV2 CYYR CYY Ratio, Y Stiffness

SDV3 CZZR CZZ Ratio, Z Stiffness

SDV4 CYZR CYZ Ratio, YZ Stiffness

SDV5 CXZR CXZ Ratio, XZ Stiffness

SDV6 CXYR CXY Ratio, XY Stiffness

SDV7 NOF Number of Failures

SDV8 CMODE Current Critical Failure Mode

SDV9 FMODE Previous Failed Mode

SDV10 CPLY Current Critical Ply Number

SDV11 FPLY Previous Failed Ply Number

SDV12 SF Safety Factor

SDV13 SUMCIJ Sum of SDV1 through SDV6

SDV14 PRODCIJ Product of SDV1 through SDV6

SDV15 MINCIJ Minimum value of SDV1 through SDV6

SDV16 MINCIJN Cij number (1 to 6) of SDV15

Abaqus

DesignationParameter Name Description

The first six parameters are stiffness ratios evaluated directly from the

elemental (global) stiffness matrix. At the beginning of the finite element analysis, the

values on the diagonal of the stiffness matrix, Cij, are stored. These values change

throughout the analysis due to material nonlinearity and progressive ply failure. The

stiffness matrix is updated and the current values occupying the diagonal are divided

by the original values to determine the stiffness ratio for that direction. Equation 4.1

shows the calculation of the CXX stiffness ratio CXXR, with the calculation of the five

other ratios following similarly.

i

XX

c

XX

C

CCXXR 4.1

67

In Equation 4.1, the c term is the current value and the i term is the initial

value. For this analysis, CXX refers to value in the (1, 1) position of the global Cij

stiffness matrix, CYY in the (2, 2), CZZ in the (3, 3), CYZ in the (4, 4), CXZ in the (5, 5),

and CXY in the (6, 6). The values of the stiffness ratios range from 1 to 0, where a

value of 1 indicates no stiffness loss from the original stiffness and 0 indicates

complete loss of stiffness of the element in the direction.

The parameter for the number of failures, NOF or SDV7, records the

number of ply failures that have occurred in the element. For example, if a [0°/90°]s

laminate had transverse tensile failure in the 90° plies and longitudinal tensile failure

in the 0° plies the number of failures would be 4. This value is a result of symmetry

with a single failure recorded in the two 90° plies and another single failure recorded

in the two 0° plies. If failure has not occurred in the element, NOF will be at zero.

This parameter does not indicate the effect of ply failure on the stiffness of the sample

but is a count of the number of failure allowables exceeded in the element up to this

point.

The critical mode CMODE, previous failed mode FMODE, critical ply

CPLY, previous failed ply FPLY, and safety factor SF are important parameters used

to determine the progressive ply failure history in LAMPATNL. The safety factor is

defined as the ratio of the principle material direction strain allowable to the current

ply level strain, as shown below:

1

1

1

TYSF if ε1>0 4.2

1

2

1

CYSF if ε1<0 4.3

68

2

3

2

TYSF if ε2>0 4.4

2

4

2

CYSF if ε2<0 4.5

3

5

3

TYSF if ε3>0 4.6

3

6

3

CYSF if ε3<0 4.7

4

7

23

YSF 4.8

5

8

13

YSF 4.9

6

9

12

YSF 4.10

where Y1T, Y1C, Y2T, Y2C, Y3T, Y3C, Y23, Y13, and Y12 are the maximum strain

allowables. These calculations are done for all N plies in the laminate. The lowest

value out of the 9N safety factors is recorded and displayed as the SF output

parameter. The ply with the lowest safety factor becomes the critical ply CPLY and

the mode in which this safety factor has occurred is recorded as the critical mode

CMODE. The values of CPLY range from 1 to N. The values of CMODE correspond

to the number of the lowest safety factor, as defined in Equations 4.2 to 4.10, and

range from 1 to 9. Critical or failure mode 1 refers to 1-direction fiber tension, mode 2

is 1-direction fiber compression, mode 3 is 2-direction matrix tension, mode 4 is 2-

direction matrix compression, mode 5 is 3-direction matrix tension, mode 6 is 3-

direction matrix compression, mode 7 is interlaminar (23) shear, mode 8 is

69

interlaminar (13) shear, and mode 9 is in-plane (12) shear. The failure modes refer to

the ply-level coordinate system that is related to the laminate and global systems as

shown in Figure 4.1.

Figure 4.1 Relation of global coordinates to laminate and ply coordinates

When a strain in a ply has reached its allowable (safety factor=1), the

failed ply and direction are discounted and excluded from the safety factor calculation.

The critical ply at this point is recorded as the previous failed ply number FPLY and

the critical mode becomes the previous failed mode FMODE. These values are

initially at zero and will take on the range of values of CPLY and CMODE if ply

failure occurs in the element. If failure is recorded, the FPLY and FMODE parameters

will portray the progression of failure in the analysis.

The SUMCIJ parameter is simply the sum of the CXXR, CYYR, CZZR,

CYZR, CXZR, and CXYR stiffness ratio parameters and can range from 6 to 0. The

PRODCIJ parameter is the product of these stiffness ratios and ranges from 1 to 0.

70

The MINCIJ parameter records the minimum value of the six stiffness ratios and also

ranges from 1 to 0. The MINCIJN parameter indicates which stiffness ratio is

recorded as the minimum for the element. If CXXR is the minimum stiffness ratio,

MINCIJN is 1, for CYYR it is 2, for CZZR it is 3, for CYZR it is 4, for CXZR it is 5,

and for CXYR it is 6. If the minimum stiffness ratio for the element is greater than

80%, no value for MINCIJN is recorded and this parameter is set to zero. Some

output parameters are not used in the following case studies but may have more

important uses in different analyses.

4.2 Design methodology

A standardized process for designing composite structures with the

LAMPATNL user material has been developed. The output parameters produced by

the user material are subsequently used to analyze the nonlinear response and

progressive failure of the composite structure. The methodology for designing a

structure using LAMPATNL is outlined in Figure 4.2.

71

Figure 4.2 Design methodology flowchart

The largest reduction in original stiffness, identified by the lowest Cij ratio

MINCIJ at the end of analysis, and its corresponding direction MINCIJN are examined

first. The next step is to determine from the NOF parameter, which records the

number of failures, whether the reduction in stiffness has been caused by material

nonlinearity or ply failures. If there are ply failures in the areas of concern, the

complete failure progression of the structure is constructed using the failed mode

72

FMODE and failed ply FPLY. If the areas of low stiffness ratio are caused by

nonlinear effects, the critical ply CPLY and mode CMODE at the final increment

become the basis for changes to the layup. The designer would use this information,

critical ply and mode or failure progression, to make changes to the design of the

structure to achieve the objectives of the analysis. Multiple iterations of this design

methodology may be needed in order to achieve these objectives. The following case

studies help to demonstrate this design methodology.

4.3 Original LAMPATNL design methodology

An example comparing the original outputs of LAMPATNL (safety factor,

critical mode, and critical ply) to the new stiffness ratios outputs is provided in this

section. This example illustrates new contributions that the stiffness ratio parameters

bring to visualizing the critical design information of the structure.

In this example, a laminate of a [0°/90°]s layup of IM7/8551-7 is tested as

a beam in bending. The Ramberg-Osgood parameters and maximum strain allowables

for this material are shown in Tables 3.4 and 3.5 respectively. The sample is

constrained in X- and Z-directions along the bottom right edge and the Z-direction

along the bottom left edge as shown in Figure 4.3. Also in Figure 4.3, a displacement

in the Z-direction along a portion of the top face is applied to introduce a bending load

into the sample.

73

Figure 4.3 Diagram of bending sample

The outputs that were originally available in LAMPATNL analysis were

the safety factor SF, critical mode CMODE, and critical ply CPLY. When a strain in a

ply has reached its allowable, the ply fails. This ply and direction are discounted and

excluded from the safety factor calculation. The recorded value of the safety factor

therefore decreases to one before jumping to a different value, as shown for the

loading of an arbitrary element in Figure 4.4.

74

0

1

2

3

4

5

6

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

Displacement (cm)

Sa

fety

Fa

cto

r

Figure 4.4 Plot of safety factor vs. displacement

At the end of the simulation for this example, the element shown in Figure

4.4 displays a safety factor of 4.53 yet there have been at least three ply failures in this

element. Information on the failure history of the laminate in this element cannot be

determined from this final value of safety factor.

Figure 4.5 shows the contour plots of safety factor from the beginning to

the end of the analysis. In these plots, blue areas have a safety factor close to 1, red

areas indicate a safety factor of 5, and grey areas are anything greater than 5. From the

plot of the final increment (last plot in the series), it can be determined that the blue

areas are close to failure, but there is no information on which areas and plies have

failed nor is there information about how these failures have affected the stiffness of

75

the structure. The complex and abrupt fluctuations in the safety factor and the

inability to determine the failure history prevent the original outputs of LAMPATNL

from effectively providing the information necessary to make changes to the

composite design.

Figure 4.5 Plot of progression of safety factor (SF)

In this work, new output parameters from LAMPATNL are proposed that

include the structural stiffness ratios of the laminate. These parameters are detailed in

the LAMPATNL output parameters section of this chapter. Figure 4.6 shows the

76

results of the MINCIJ parameter in the same arbitrary element documented in Figure

4.4. This plot shows that there was catastrophic stiffness loss in at least one direction

for this element. At the end of the analysis, the value of this parameter is 0.007,

indicating that complete stiffness loss has occurred in at least one direction in this

element.

0

0.2

0.4

0.6

0.8

1

1.2

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

Displacement (cm)

Min

imu

m S

tiff

ne

ss

Rati

o

Figure 4.6 Plot of minimum stiffness ratio (MINCIJ) vs. displacement

A progression plot of the MINCIJ parameter is shown in Figure 4.7. In

this plot, areas that are blue correspond to a zero stiffness ratio (complete stiffness

loss) and areas that are red correspond to no stiffness loss. By examining the contour

77

plot of the last increment, it is easy to determine which areas have severe stiffness loss

and which areas do not. Because of the simple nature of the stiffness ratio measure, it

is easy to determine from the progression which areas have failed and in which order

they have failed. This contrasts from the safety factor plots shown in Figure 4.5,

which do not efficiently visualize the failure history of the part.

Figure 4.7 Plot of progression of minimum stiffness ratio (MINCIJ)

78

4.4 Case study #1

4.4.1 Description

The first case study that is examined with the LAMPATNL is a

compressive shear sample. This sample was created to represent a composite material

that is bonded to a much stiffer material. The sample, shown in Figure 4.8, is

rectangular with an aspect ratio of 2.5 that is fixed on one half of one side. A

compressive load in the X-direction is applied to one end of the sample and a

boundary condition on the opposing end prevents the sample from moving in the X-

direction.

79

Figure 4.8 Diagram of shear sample

The first design tested is a [0°/90°]s layup of AS4/3501-6. The Ramberg-

Osgood parameters and maximum strain allowables for this material are shown in

Tables 3.4 and 3.5, respectively. A uniform pressure of 800 MPa is applied to the top

80

end of the sample, causing compressive stresses and a shear stress concentration close

to the right fixed boundary condition.

The sample is modeled using a mesh of 6448 three-dimensional 8-node

linear brick elements with reduced integration, type C3D8R in ABAQUS. The mesh

contains 8505 nodes each with three degrees of freedom, resulting in 25515 total

degrees of freedom. For each case study, the mesh was varied in both density and

structure to ensure the results of the simulation did not change dramatically.

The objective of this case study is to limit the displacement of point A (see

Figure 4.8) in the X-direction to 0.05 m. Limiting the shear and longitudinal failure is

important to the performance of this sample, so reducing areas of stiffness loss in these

directions is also an objective. Based upon the performance of the first layup,

modifications to the laminate may be needed to achieve the objective. Any change to

the laminate must maintain the same material, weight, and thickness from the original

layup. These changes are made based on information gained from the process outlined

in the design methodology.

4.4.2 Results of original design

Following the procedure outlined in the design methodology, the first step

is to investigate the lowest ratio of current stiffness to original stiffness at the end of

the analysis. The contour plot of the LAMPATNL output parameter MINCIJ is shown

in Figure 4.9. In this plot, areas of concern are those that are blue, corresponding to a

zero stiffness ratio, and areas of least concern are red, corresponding to no stiffness

loss. This color scale holds for all plots of the MINCIJ, CXXR, CYYR, CZZR, and

CXYR shown in this case study.

81

Figure 4.9 Contour plot of minimum Cij stiffness ratio, MINCIJ

82

A significant area of the structure has at least one of its stiffness ratios

near zero. The next step in this methodology is to determine which direction the

stiffness ratios are at or near zero. A plot of the Cij number (1-6), MINCIJN, indicates

the critical direction of the lowest stiffness ratio and is shown in Figure 4.10.

Figure 4.10 Contour plot of the direction of minimum Cij ratio, MINCIJN

83

The area in the bottom right portion of the structure at the fixed half-face

has CXY (red area of MINCIJN=6) as the lowest stiffness ratio, corresponding to the

XY shear stiffness. The areas in green also have a stiffness ratio close to zero and

correspond to a value of 3 in this plot. This result shows that there is a loss of stiffness

in the CZZ stiffness ratio, or global Z-direction. This is not a result of loading in the Z-

direction, but a result of ply strains caused by Poisson’s effects exceeding the low

through-thickness tensile failure allowable.

In order to further investigate the critical losses in stiffness, a closer

examination of each individual stiffness ratio is required. Upon inspection, the

CXXR, CYYR, CZZR, and CXYR stiffness ratio parameters are low in the areas of

interest. The contour plot of the CZZR ratio in Figure 4.11 confirms what is shown in

Figure 4.10, that a large part of the sample has lost stiffness in the Z-direction. The

plot of the CXY stiffness ratio, shown in Figure 4.12, also confirms what is shown in

Figure 4.10, that the XY shear stiffness is reduced near the fixed face of the sample.

84

Figure 4.11 Plot of CZZ stiffness ratio, CZZR

85

Figure 4.12 Plot of CXY stiffness ratio, CXYR

A closer examination of the plot of the CYY stiffness ratio shows that the

upper portion of the fixed face also experiences a loss of stiffness in the Y-direction as

86

shown in Figure 4.13. The limitation of the contour plot in Figure 4.10 is that only

one value for each element can be displayed. In the upper portion of the fixed face, the

loss of stiffness occurs in both the Y- and XY-directions although only the shear

direction is indicated in this area in Figure 4.10.

Figure 4.13 Plot of CYY stiffness ratio, CYYR

87

A similar scenario occurs in the areas of Figure 4.10 that indicate Z-

direction failure. Figure 4.14 shows the plot of the CXX stiffness ratio, in which a large

part of the area that had stiffness loss in the Z-direction also has a loss of X-direction

stiffness. Because the sample is loaded in this direction, this failure is important to the

response of the structure.

Figure 4.14 Plot of CXX stiffness ratio, CXXR

88

The next step of the design methodology is to determine if there are any

ply failures recorded. Because a large part of the sample has a stiffness ratio at or near

zero, ply failures are to be expected. A plot of the number of failures that have

occurred is shown in Figure 4.15. The upper portion of the fixed face has the most ply

failures. This plot also shows areas that have a loss of stiffness in the X-direction are

differentiated from those with only Z-direction failure, shown as the areas of orange

and green, respectively.

Figure 4.15 Contour plot of number of failures, NOF

89

Continuing in the design methodology, the ply-level failure history of the

sample is investigated. Ply-level failure behavior is the cause of the global stiffness

loss shown in the stiffness ratio plots in Figures 4.11 to 4.14. Figure 4.16 displays the

progression of the modes that are failing throughout the response of the structure. The

failure modes shown in this figure refer to the ply-level coordinate system as described

in the LAMPATNL output parameters section of this chapter.

In Figure 4.16, only the mode of failure for the last failed ply is displayed.

Areas in dark blue (FMODE=0) have not failed, red (FMODE=9) indicates a failure

mode of 12 shear, green areas (FMODE=5) have tensile failure in the 3-direction, the

areas in pale green (FMODE=3) are 2-direction tensile failure, and the light blue areas

(FMODE=2) indicate 1-direction compressive failure. It is important to understand

which of the four plies correspond to the failure modes of Figure 4.16. A progression

plot of the last failed ply is shown in Figure 4.17.

In Figure 4.17, the blue areas indicate that no failure has occurred, red

areas (FPLY=4) indicate that the failure shown in Figure 4.16 occurred in the 0° plies,

and the light orange area (FPLY=3) indicates that the failure occurred in the 90° plies.

In order to describe the progression of failure in the sample, four regions are identified

and highlighted on the sample below in Figure 4.18. The regions, shown in Figure

4.18 on a plot of FMODE at the end of the analysis, are used to refer to areas in the

progression plots of Figure 4.16a-e and Figure 4.17a-e.

Figure 4.16 Plot of progression of last failed mode, FMODE

90

Figure 4.17 Plot of progression of last failed ply, FPLY

91

92

Figure 4.18 Diagram of regions

There are three phases of failure that occur in this sample. The first phase

is in-plane shear failure in all of the plies that propagates from region 1, the top edge

of the fixed face, downwards into region 2. This phase is displayed in plots a and b of

Figures 4.16 and 4.17. The next phase includes tensile failure in the 3-direction in all

plies that starts at the top of the fixed face (region 1) and continues up the right edge of

the sample (region 3). This phase is shown in plots c and d of Figures 4.16 and 4.17.

93

Additional tensile failure in the 3-direction occurs on the face opposing the fixed

portion, shown in Region 4 of plots c and d. The final phase is compressive 1-

direction failure in the 0° plies starting in Region 1 of plot c, expanding to Region 3 in

plots d and e, and also occurring in Region 4 in plots d and e.

Figures 4.9 through 4.17 display the information that is needed to gain

insight into the ply-level failure behavior of the sample. This insight is used to make

changes to the layup of the sample to meet the objectives of this case study. The

current [0°/90°]s layup does not satisfy both of the objectives of the case study. The

displacement at point A is 0.0585 m, which exceeds the target value of 0.05 m. The

areas of global XY shear failure (Figure 4.12) and global X-direction failure (Figure

4.14) are large and limiting these areas is the other important objective.

4.4.3 Design iteration #1

The original [0°/90°]s layup of the laminate is insufficient in both the

displacement objective and limiting the extent of stiffness loss. The design is first

changed to a [02/±45°]s layup. The 0° plies of the laminate are essential in providing

enough longitudinal stiffness to satisfy the displacement objective. Using ±45° plies is

aimed at addressing the area of shear failure along the right fixed boundary condition.

The fiber direction of these plies is aligned more in the global X-direction than the 90°

plies, adding to the stiffness of the structure in this direction. In the original design,

the 90° plies only provided minimal stiffness in the X-direction.

In order to determine if the new layup has improved upon the original

design, the stiffness ratios are first examined. In Figure 4.19, the MINCIJ of the new

design is compared to the original. The new design does not have areas with a

complete loss in stiffness in any direction (dark blue areas). The new design has only

94

a small area in the lower right corner where a minimum stiffness ratio has not changed

(red area) from its original value. In the original design, the area of no loss of original

stiffness is large.

Figure 4.19 Comparison of minimum stiffness ratio MINCIJ [0°/90°]s (left) vs.

[02/±45°]s (right)

The majority of the sample in right plot of Figure 4.19 has some stiffness

loss. The plot shown in Figure 4.20 is used to determine in which direction these

95

losses are the greatest. In this contour plot, the areas in red signify the Y-direction

(CYYR) and the areas in green signify the X-direction (CXXR), the blue area indicates

that the minimum stiffness ratio was not below 80%. While the stiffness in the X-

direction is critical to the response of the structure, stiffness loss in the Y-direction is

not as critical because there is no loading in this direction though there will be a

Poisson’s effect.

Figure 4.20 Plot of the direction of minimum Cij ratio MINCIJN for [02/±45]s

layup

96

A comparison of the individual stiffness ratios of CXX, CYY, CZZ, and CXY

will determine the benefits, if any, of switching to this layup. The comparison plot of

CXX in Figure 4.21 shows that while the stiffness loss in the X-direction of the

[02/±45]s sample covers larger area than the original design, the magnitude of stiffness

loss in these areas is less. In this plot, the areas in light blue have a higher stiffness

ratio than those in dark blue.

Figure 4.21 Plot of CXX stiffness ratio CXXR comparing [0°/90°]s (left) to

[02/±45]s (right)

97

Comparing the CYY stiffness ratios of the sample in Figure 4.22, it is seen

that the new layup has a large area where stiffness has been reduced. This reduced

stiffness is a result of the 0° plies failing in the transverse matrix direction, which

causes a small reduction in stiffness in this area.

Figure 4.22 Plot of CYY stiffness ratio (CYYR) comparing [0°/90°]s (left) to

[02/±45]s (right)

98

Similarly, the comparison of CZZ ratio shown in Figure 4.23 illustrates that

the complete failure of some elements in the Z-direction has been replaced by slight

softening over the entire structure. The response of the sample in this direction is

determined by the global XZ Poisson ratio and the new layup has less coupling in this

direction.

Figure 4.23 Plot of CZZ stiffness ratio (CZZR) comparing [0°/90°]s (left) to

[02/±45]s (right)

99

Finally, an examination of the global XY shear stiffness ratio between the

original and modified layups is shown in Figure 4.24. The modified layup shows

dramatic improvements in limiting the extent and severity of shear failure along the

fixed boundary.

Figure 4.24 Plot of CXY stiffness ratio (CXYR) comparing [0°/90°]s (left) to

[02/±45]s (right)

100

The results of this first design iteration are a reduction in the area and

magnitude of XY shear stiffness loss, a reduction in magnitude of Z stiffness loss,

more widespread but less severe Y stiffness loss, and a decrease in the magnitude of

stiffness loss in the X-direction. The displacement of point A for this layup is

however, 0.0647 m; which is worse than the first layup tested. While there is

improved performance in the stiffness objectives, the displacement objective is still

not met for this layup.

4.4.4 Design iteration #2

For the next design iteration, the stiffness in the longitudinal direction of

the structure must be improved in order to meet the displacement objective. A

laminate in a [02/±35°]s layup is chosen because the ±35° plies are needed to address

the XY shear failures that occur along the fixed boundary of the part. These plies also

increase the global X-direction stiffness of the sample since their fibers are oriented

more in this direction than the previous design.

The first condition to check is the displacement objective. Examining the

results of this sample shows that the X displacement at point A is once again not

within the desired objective, at 0.0585 m. An investigation into the stiffness ratio

parameters shows that there is an improved performance in the X stiffness, slightly

less desirable performance in the Y and Z directions, and similar performance in the

XY shear stiffness when compared to the first design iteration.

101

4.4.5 Design iteration #3

The next layup needs to increase the stiffness of the laminate in the X-

direction while continuing to prevent shear stiffness loss. A laminate in a [02/±25°]s

layup satisfies both these criteria and is used for this design iteration.

The first check for the layup is to compare the displacement at point A

to the targeted value. A displacement of 0.0475 m, less than the targeted value, is

recorded at this point. With this objective satisfied, an examination of the four key

stiffness ratios is needed to confirm that the performance of the layup meets the other

objective.

Like the first design iteration, an examination of the MINCIJ of this layup

is compared to the original (see Figure 4.25). The majority of the sample with the new

layup has a significant loss in stiffness in at least one direction. However, in contrast

with the original layup, this layup does not have large areas of complete stiffness loss.

The new layup has only one element, near the top of the fixed edge, with complete

stiffness loss.

102

Figure 4.25 Comparison of minimum stiffness ratio MINCIJ, [0°/90°]s (left) vs.

[02/±25°]s (right)

The plot shown in Figure 4.26 displays that the Y-direction, here shown in

pale green, is the predominate loss of stiffness direction. Stiffness loss in the Y-

direction is not critical because there is no loading in this direction.

103

Figure 4.26 Plot of the direction of minimum Cij ratio MINCIJN for [02/±25]s

layup

104

A comparison of the individual stiffness ratios of CXX, CYY, CZZ, and CXY

is once again performed. The comparison plot of CXX in Figure 4.27 shows that the

stiffness loss in the X-direction has been minimized. In this plot, the areas in blue have

a high stiffness loss and the areas in red have no stiffness loss.

Figure 4.27 Plot of CXX stiffness ratio (CXXR) comparing [0°/90°]s (left) to

[02/±25]s (right)

105

Comparing the CYY stiffness of the sample in Figure 4.28, it is seen that

the new layup has a large area where stiffness has been reduced. Because stiffness in

this direction is not critical to the overall response of the sample, this result is not a

concern.

Figure 4.28 Plot of CYY stiffness ratio (CYYR) comparing [0°/90°]s (left) to

[02/±25]s (right)

106

Similar to the first design iteration, the comparison of CZZ shown in Figure

4.29 illustrates that the complete failure of some elements in the Z-direction has been

replaced by slight softening over the entire structure.

Figure 4.29 Plot of CZZ stiffness ratio (CZZR) comparing [0°/90°]s (left) to

[02/±25]s (right)

107

Finally, an examination of the XY shear stiffness ratio between the

original and modified layups is shown in Figure 4.30. The modified layup shows

improvements in limiting the extent and severity of shear failure along the fixed

boundary when compared to the original layup. This layup does not perform as well as

the first design iteration, but it does eliminate the shear failure region.

Figure 4.30 Plot of CXY stiffness ratio (CXYR) comparing [0°/90°]s (left) to

[02/±25]s (right)

108

This layup is able to satisfy both objectives set at the start of the case

study. With a displacement of 0.0475 at point A, the sample is within the targeted

value of 0.05. The modifications to the layup limit the areas of total stiffness loss of

the structure in the global X and XY shear directions.

This case study demonstrates the utility of the stiffness ratio, last failed

mode, and last failed ply LAMPATNL output parameters. These parameters helped to

improve the understanding of the complex interactions between progressive ply failure

and global stiffness reduction in the laminate. Areas of critical stiffness loss were

easily identified using the linear scale of the minimum stiffness ratio parameter. The

progression of ply failures, determined from the design methodology, provided a clear

understanding of the response of the structure. Modifications to the [0°/90°]s layup

were made based on the insight gained from the mode and ply failure history. These

modifications enabled the designer to quickly achieve the performance objectives of

the analysis using only three design iterations.

4.5 Case study #2

4.5.1 Description

The second case study is three dimensional model of an open hole sample

that is subjected to multiaxial, in-plane loading. The open hole configuration was

selected because it is a standard example used to examine progressive failure models

for composites [18, 19, 21-25, 27, 29, 30]. This problem is also common in composite

applications and the effects on the structural response due to changes in material,

layup, and width-to-diameter ratio are important to understand. A diagram of the

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sample is shown in Figure 4.31. The diameter of the hole is 1 and the width of the

sample in both the X- and Y-directions is 5, resulting in width-to-diameter ratio of 5.

Figure 4.31 Diagram of open hole sample

The original laminate is an S-Glass/Epoxy material in a [0°/90°/±45°]s, or

quasi-isotropic, layup. The Ramberg-Osgood parameters and maximum strain

allowables for this material are shown in Tables 3.4 and 3.5, respectively [2, 36]. The

mesh for this model contains 6768 elements and 8905 nodes. The finite element

model has symmetry boundary conditions along the left face and bottom face. A

uniform pressure load of 300 MPa is applied to the right exterior faces in the positive

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X direction, creating tensile stresses in that direction. Another pressure load of 150

MPa is applied to the top exterior faces in the negative Y direction, creating

compressive stresses in that direction.

The objectives in this design case study are to limit the displacement of

the point at the upper right corner of the structure to less than 0.05 m in the positive X-

direction and 0.04 m in the negative Y-direction. Based upon the performance of the

quasi-isotropic layup, modifications to the laminate may be needed to achieve these

objectives. Any change to the laminate must maintain the same material, weight, and

thickness from the original layup. These changes are made based on information

gained from the process outlined in the design methodology.

4.5.2 Results of original design

Following the procedure of the design methodology, the first contour plot

examined is a plot of the minimum Cij ratio MINCIJ, shown in Figure 4.32.

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Figure 4.32 Contour plot of minimum Cij stiffness ratio MINCIJ

The yellow and bright green areas of the sample represent a stiffness ratio

of around 60%-70%, the pale blue area has a stiffness ratio of 10%-20%, the dark blue

area has 0% stiffness ratio, and the red area (2 elements) is near 100% stiffness ratio.

The multiaxial loading that is applied to this sample causes stress

concentrations at both sides of the open hole and, as expected, the stiffness ratio is

lowest in these areas. Almost the entire sample has a 30% reduction of stiffness in at

least one direction. To determine which directions are represented in Figure 4.32, a

plot of the minimum stiffness ratio number MINCIJN is needed. This plot is shown in

Figure 4.33.

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Figure 4.33 Contour plot of the direction of minimum Cij ratio MINCIJN

In this figure, light blue (MINCIJN=1) represents the X-direction, pale

green (MINCIJN=2) represents the Y-direction, and bright green (MINCIJN=3)

represents the Z-direction. The important information from this plot is that the area of

stiffness ratio close to 10% at the top of the hole corresponds to the global X-direction

and the area of complete loss in stiffness to the right of the hole corresponds to the Z-

direction. The rest of the sample that has a 30% reduction in stiffness is a combination

of the X- and Y-directions.

The reduction of stiffness at the top and right of the hole are an indication

that failure probably has occurred in these areas. The contour plot of the number of

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failures in Figure 4.34 shows that these areas have the highest amount of ply failures in

the sample. As a result, an investigation into all of the critical stiffness ratios is

needed because the stiffness reduction may have occurred in multiple directions in

addition to what is indicated in Figures 4.32 and 4.33.

Figure 4.34 Contour plot of number of failures, NOF

The multiaxial loading combined with XZ and YZ Poisson ratio effects

contributes to strains in the Z-direction of the sample. The softening and failure of the

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laminate in the through-thickness direction is shown in Figure 4.35. The majority of

this sample has a 15% reduction in stiffness, the area above the open hole has been

reduced by 35%, and the area to the right of the hole has a complete loss in stiffness.

This stiffness is not important to the overall response of the sample, but its

contribution to the minimum stiffness ratio shown in Figures 4.32 and 4.33 could

mask the reduction in other important stiffness ratios.

Figure 4.35 Plot of CZZ stiffness ratio (CZZR)

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The next stiffness ratio examined is the CXX ratio that is shown in Figure

4.36. The three areas to note in this plot are the top of the hole, the right of the hole,

and the remainder of the sample. At the top of the hole, the CXX stiffness has been

reduced to 15% of its original value. To the right of the hole and in the rest of the

sample, this stiffness ranges from 50% to 75% of its original value. It is important to

note that in the area to the right of the hole, this stiffness ratio has only been reduced to

50% of its original stiffness. This suggests that, although some plies in this region

may have failed in the global X-direction, the 0° plies in this direction have not failed

in fiber tension. The ply failure history of this sample is documented later in this

section.

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Figure 4.36 Plot of CXX stiffness ratio (CXXR)

The plot of the CYY stiffness ratio in Figure 4.37 has similar characteristics

to the previous plot. At the right of the hole, the CYY stiffness has been reduced to

15% of its original value. This significant drop in stiffness is not apparent when

looking at the minimum stiffness ratio plots. To the top of the hole and in the

remainder of the sample, this stiffness ranges between 50% and 75% of its original

value.

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Figure 4.37 Plot of CYY stiffness ratio (CYYR)

The final stiffness ratio that is important to the response of the structure is

the CXY stiffness ratio, which represents the XY shear stiffness. In Figure 4.38, the

majority of the sample has less than 5% stiffness loss in this direction. Areas in close

proximity to the hole in the sample have been reduced to 65% of the original shear

stiffness.

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Figure 4.38 Plot of CXY stiffness ratio (CXYR)

The progression of failure through the sample must be understood before

making changes to the layup. A progression plot of the last failed mode is shown in

Figure 4.39. For this plot, dark blue indicates no failure (FMODE=0), the blue areas

(FMODE=1) indicate 1-direction tensile failure, the light blue areas (FMODE=2)

indicate 1-direction compressive failure, the areas in pale green (FMODE=3) are 2-

direction tensile failure, bright green areas (FMODE=5) have tensile failure in the 3-

direction, and red (FMODE=9) indicates a failure mode of 12 shear.

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A progression plot of the last failed ply is shown in Figure 4.40. In this

figure, grey areas indicate no failure recorded (FPLY=0), yellow areas refer to the 90°

plies (FPLY=7), red areas to the 0° plies (FPLY=8), blue areas to the -45° plies

(FPLY=5), and finally pale green areas refer to +45° plies (FPLY=6).

Figure 4.39 Plot of progression of last failed mode FMODE

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Figure 4.40 Plot of progression of last failed ply FPLY

121

122

The quasi-isotropic layup of the laminate makes the interpretation of these

plots more difficult than the first case study. In order to better describe the progression

of failure in the sample, five regions are identified and highlighted on the sample

below in Figure 4.41. The regions, shown in Figure 4.41 on a plot of FMODE at the

end of the analysis, are used to refer to areas in the progression plots of Figure 4.39a-g

and Figure 4.40a-g.

Figure 4.41 Diagram of regions

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Failure begins in this sample in region 1, the top portion of the hole, with

transverse (2-direction) tensile failure in the 90° plies (see plot a of Figures 4.39 and

4.40) and propagates across almost the entire sample, shown in the yellow areas of

Figure 4.40 plots b and c. The next failures again start in region 1 during plot b and

are transverse tensile failure in the ±45° plies. These failures expand outward to

region 2 throughout the rest of the loading as shown in the green and blue areas in

regions 1 and 2 of Figure 4.40 plots c to g. An area of tensile 3-direction failure in all

plies starts in region 3, to the right of the hole, in plot c and expands to encompass this

region from plots d to g. Also in this region, both compressive transverse failure in the

0° plies and tensile transverse failure in the 90° plies occur from plots d to g. A pocket

of transverse tensile failure in the ±45° plies (blue area in region 4 of Figure 4.40e),

followed by compressive transverse failure in the 0° plies (red area in region 4 of

Figure 4.40f), arises and expands in region 4. These areas of failure expand into

region 5 through plots f and g. Finally, 12 shear failure occurs in the ±45° plies, which

is the red area in region 4 of Figure 4.39g.

The quasi-isotropic layup results in a displacement of 0.074 m in the

positive X-direction and 0.046 m in the negative Y-direction. These values are not

within the desired displacement objective in both directions, so changes to the layup

must be made to address these problems. In addition, there is a large area of failure in

this sample and minimizing this area is essential in meeting the displacement

objective.

4.5.3 Design iteration #1

Any changes to the original layup must simultaneously improve stiffness

in the global X- and Y-directions, because each displacement objective has not been

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met. To achieve this, no additional plies can be added to the new layup. In the

[0°/90°/±45°]s layup, the 0° plies are essential to the response in the X-direction as are

the 90° plies for the response to the Y-direction. Changing the remaining ±45° plies to

additional 0° and 90° plies would help achieve the displacement objectives, but there

is XY shear failure already in the sample and losing the cross-plies would increase this

failure. A change to the ±45° is needed to address either the X-direction or the Y-

direction. Since the displacement in the X-direction is farther from its objective,

orienting the fibers more in this direction will be beneficial.

A [0°/90°/±30°]s layup is tested in this design iteration. A check of the

displacements at the top right corner of the sample shows improvements in both

directions. The displacement in the positive X-direction is 0.052, greater than the

targeted value. The displacement in the negative Y-direction is 0.0405, slightly greater

than the targeted value.

This analysis shows a decrease in the magnitude of the displacement in the

Y-direction. This result is unexpected because cross-ply fibers are oriented less in this

direction when compared to the original layup, yet the displacement in this direction

has decreased. In the original analysis, there were several areas of the sample that had

transverse tensile failure in the ±45° plies. These failures were caused by the tensile

loading in the X-direction of the sample and result in the 2-direction stiffness in these

plies to be discounted. The overall stiffness of the structure in the Y-direction is

softened, causing more deflection.

A comparison between the original layup and the current layup of the CYY

stiffness ratios at the end of loading is shown in Figure 4.42. The initial stiffness of

the new layup in the Y-direction is less than that of the original layup, but the stiffness

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ratio at the end of loading is greater. The area at the top of the open hole retains 30%

more stiffness when compared to original layup. The bulk of the sample has a

stiffness ratio ranging from 70% to 95%, compared to 50% to 75% in the original

layup. The area to the right of the hole (region 3) exhibits more stiffness loss than the

original layup, a 5% ratio compared to a 15% ratio.

Figure 4.42 Plot of CYY stiffness ratio (CYYR) comparing [0°/90°/±45°]s (left) to

[0°/90°/±30°]s (right)

The prevention of stiffness loss in the new layup comes from limiting the

transverse tensile failure in the ±30° plies, illustrated below in Figure 4.43. The plot

shows the last failed ply at the end of loading. The areas in dark blue and pale green

represent cross-plies that have failed. The corresponding failure mode in these areas is

transverse (2-direction) tensile failure. In the left plot of Figure 4.43, the transverse

cross-ply failure region is large, expanding across the lower portion of the sample

(region 5) and up from the top of the hole (regions 1 and 2). In the right plot of Figure

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4.43, the area of cross-ply failure is only a small portion on the top edge of the hole

(region 1), significantly smaller than the original layup.

Figure 4.43 Plot of last failed ply (FPLY) comparing [0°/90°/±45°]s (left) to

[0°/90°/±30°]s (right)

4.5.4 Design iteration #2

Building on the results and insights of the first design iteration, additional

changes need to be made to the layup to meet the desired performance objectives. The

displacement of the corner point in the negative Y-direction must be below 0.04 m, so

improving the stiffness and reducing progressive failure in this direction is once again

needed. Based on the first iteration, aligning the fibers further into the X-direction

will prevent transverse tensile failure in these plies, minimizing stiffness loss in the Y-

direction, and decreasing the magnitude of the Y-displacement.

The layup used for this design iteration is [0°/90°/±20°]s, which increases

the fiber orientation of the plus/minus plies by 10° over the previous iteration. The

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resulting displacements of the upper right corner of the sample are 0.041 m in the

positive X-direction and 0.0342 m in the negative Y-direction. Both displacements are

within their respective target values, so this layup has achieved the performance

objectives. Comparisons of the critical stiffness ratios will display the improvements

in reducing the progressive failure within the structure.

The comparison of the minimum Cij stiffness ratio MINCIJ is shown in

Figure 4.44. When compared to the original layup, the improvements are not

apparent. The light blue area representing a 15% stiffness ratio in the new layup

(Figure 4.44 right) is smaller and confined close the edge of the hole (region 1). The

area of complete stiffness loss to the right of the hole (region 3) is larger in the new

layup. Throughout the rest of the sample, the minimum stiffness ratios are greater than

that of the original design.

Figure 4.44 Comparison of minimum stiffness ratio (MINCIJ), [0°/90°/±45°]s

(left) to [0°/90°/±20°]s (right)

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A contour plot of the direction of minimum Cij stiffness ratio MINCIJN is

shown in Figure 4.45. The areas in red signify the XY shear ratio, pale green signify

the Y-direction, light blue (1 element) is the X-direction, and bright green is the Z-

direction. Dark blue areas have a minimum stiffness ratio greater than 80% and the

direction is not recorded.

Figure 4.45 Plot of the direction of minimum Cij ratio MINCIJN for

[0°/90°/±20°]s layup

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The plot of CXX stiffness ratio in Figure 4.46 shows the improved

performance of the new layup in this direction. With the ±20° fibers aligned more in

the X-direction than the quasi-isotropic layup, the new layup is expected to have less

softening and failure in this direction. The area of 85% stiffness loss in region 1, the

top of the hole, for the original layup has been reduced to one element along the edge

of the hole in the new layup. The rest of the sample has become just 5% to 25%

compliant, an improvement on the 25% to 40% compliance of the original layup.

Figure 4.46 Plot of CXX stiffness ratio (CXXR) comparing [0°/90°/±45°]s (left) to

[0°/90°/±20°]s (right)

Like the first design iteration, the CYY stiffness ratios shown in Figure

4.47 improve upon the original layup. Limiting the transverse tensile failure in the

±20° plies improves the performance of the sample in the Y-direction. The area of

stiffness loss around the right side of the hole (region 3) is slightly more severe than

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the original layup, but the improvement in stiffness loss over the entire sample

outweighs this drawback.

Figure 4.47 Plot of CYY stiffness ratio (CYYR) comparing [0°/90°/±45°]s (left) to

[0°/90°/±20°]s (right)

The CZZ stiffness ratio plot in Figure 4.48 shows improvement from the

original layup throughout the entire structure. The area of complete stiffness loss in

right plot of Figure 4.44 is shown here as stiffness loss in the Z-direction. The XZ and

YZ coupling of this material causes strain in the Z-direction, but this is not a critical

direction because there is no loading in this direction.

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Figure 4.48 Plot of CZZ stiffness ratio (CZZR) comparing [0°/90°/±45°]s (left) to

[0°/90°/±20°]s (right)

The one aspect where stiffness loss has increased from the original design

to the new design is the XY shear stiffness. The plot of the CXY stiffness ratio shown

in Figure 4.49 illustrates the extent of shear stiffness loss. While the original layup

had very little area of concern, the new design has a 15% stiffness ratio around the

upper edge of the hole (region 1) and a stiffness ratio between 50% to 85% throughout

the rest of the structure.

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Figure 4.49 Plot of CXY stiffness ratio (CXYR) comparing [0°/90°/±45°]s (left) to

[0°/90°/±20°]s (right)

The [0°/90°/±20°]s layup achieves the displacement performance

objectives of the analysis. These objectives are achieved by increasing the stiffness of

the structure in the X-direction and limiting the amount of transverse tensile failure in

the plus/minus plies of the layup. Comparisons of the stiffness ratios between the

original and new layup display improvements in reducing the amount of stiffness loss

in the sample.

This case study shows that the approach outlined in the design

methodology improves the understanding of the complex material nonlinearity and

progressive ply failure response of the open hole sample. The stiffness ratio

parameters provide a straightforward measure of material degradation and reduce the

complexity of the results when compared to the safety factor output parameter of the

original LAMPATNL program. Areas of critical stiffness loss were easily identified

using the minimum stiffness ratio parameter. The failed mode and ply plots for this

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sample identified transverse tensile failure in the ±45° plies of the quasi-isotropic

layup as a cause of stiffness loss in the global Y-direction. Modifications to the

[0°/90°/±45°]s layup were made to address this failure. These modifications enabled

the designer to quickly achieve the performance objectives of the analysis.

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Chapter 5

CONCLUSIONS

5.1 Conclusions

The development of a methodology to design and analyze thick section

composite structures using nonlinear ply properties, progressive ply failure, and a

homogenizing-dehomogenizing technique (the “smearing-unsmearing” approach) was

the goal of this research. An investigation into the ability of ABAQUS, ANSYS, and

MSC.Nastran to model composite materials with anisotropic nonlinear behavior,

support progressive ply failure using multiple failure criteria, homogenize laminates to

obtain effective properties, and dehomogenize the laminate to apply ply based failure

theories was conducted. The results of this investigation showed that none of the

programs contained all of these abilities. Additional composite modeling software,

such as GENOA and Heilus:MCT, were also researched and yielded similar

conclusions.

Presented in this work was background on the various capabilities and

limitations of the “LAM” series of codes, LAM3D, LAM3DNLP, LAMPAT, and

LAMPATNL. These codes are able to model thick section composite structures using

nonlinear ply properties, progressive ply failure, and the “smearing-unsmearing”

approach. In the “LAM” codes, ply-level material nonlinearity is represented by using

the Ramberg-Osgood equation. Progressive ply failure uses the ply discount method

in combinations with ply-based failure criteria. The “smearing-unsmearing” approach

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uses the theory and assumptions of Chou [37] to calculate laminate and ply-level

stress-strain response.

Out of these codes, LAMPATNL had been integrated into ABAQUS using

a user material subroutine. LAMPATNL is the finite element implementation of the

nonlinear, analytical model LAM3DNLP. There were several modifications to the

procedure of LAM3DNLP in order for it to function in the finite element environment

as LAMPATNL. The modifications include changes to the procedure for calculating

ply-level stresses and strains, continuously updating of the Poisson’s ratios to satisfy

material stability conditions, and changes to the post-failure behavior of the code.

Until the discussions in this research, these changes were never documented and their

implications were never discussed.

The original LAMPATNL user material was not able to accurately track

the failure history of the elements of the structure. Also, support of elements was

limited to axisymmetric and plane strain types. A major contribution from this

research was the addition of the ability to track the failure history by element number,

integration point, ply, and direction. LAMPATNL and the other “LAM” codes were

developed to analyze thick section composite structures, so three-dimensional analysis

capabilities were needed. Another contribution of this research was expanding the

user material to support three-dimensional 8-node linear brick elements.

In addition to the undocumented changes to LAM3DNLP when it was

implemented in LAMPATNL, a full validation of the stress-strain response output of

this user material was also never provided. With four materials and six different

layups, a total of 13 load cases were simulated using LAMPATNL and compared to

results generated by LAM3DNLP. Both linear and nonlinear ply properties were

136

tested for these load cases to ensure full compatibility with both LAM3D and

LAM3DNLP. None of the example cases presented for validation of LAMPATNL

deviate from the values produced by LAM3DNLP.

5.2 Summary of results

LAMPATNL had the original output parameters of safety factor, critical

mode, and critical ply. It was extremely difficult to determine the areas and directions

of failure in the structure at the end of analysis using these parameters alone. An

example was provided to illustrate the difficulty of using the safety factor to analyze a

composite structure. The most important contribution of this research was the

stiffness ratio parameters output by the UMAT. These parameters improved the

visualization of progressive failure and stiffness loss and significantly increased the

utility of LAMPATNL.

The validated LAMPATNL user material subroutine was then used to

analyze thick section composite structures. Another unique contribution of this

research was the newly developed design methodology, which outlined the process of

designing a structure using the UMAT. The methodology used stiffness ratios and

failure progression to provide insight into the failure mechanics of the laminate.

Changes to the layup of the composite were made based upon the objectives of the

analysis using the information provided by the design methodology.

Two example cases were used to evaluate the process of the design

methodology. The first case study was a [0°/90°]s layup of AS4/3501-6 subjected to

compression and in-plane shearing. The objective of this case was to limit the

displacement in the X-direction. This objective was achieved by increasing the

orientation of fibers in X-direction and limiting failure in the sample. The resulting

137

design for this case was a [02/±25°]s layup. This layup was able to achieve the

performance objective of the case study by limiting the amounts of shear and tensile

failure and increasing the longitudinal stiffness in the sample.

The second case study was an open hole sample of S-Glass/Epoxy in a

quasi-isotropic layup subjected to multiaxial in-plane loading. The failure progression

of this structure revealed that transverse tensile failure in the ±45° plies resulted in

decreased stiffness in the Y-direction. Increasing the orientation of the plus/minus

plies into the X-direction leads to improved stiffness in both directions due to the

limiting of transverse tensile failure in these plies. A [0°/90°/±20°]s layup was

determined to meet both objectives of the analysis while keeping the ply amount

constant and therefore not adding weight to the structure.

5.3 Future work

The documentation and validation of the LAMPATNL user material

subroutine provides a basis to start analyzing thick section composite structures. The

design methodology provides a template for using LAMPATNL to modify nonlinear

composite layups using progressive ply failure analysis. The two case studies

illustrated the application of the user material to design the layup of two structures

under complex loading. The expansion of the LAMPATNL to analyze other cases of

thick section composite structures is one subject of future work. In addition to the

design and analysis of these structures, experimental validation of the resulting design

modifications from LAMPATNL is also an important step.

Another subject of future work from this research is expanding the

LAMPATNL user material to explicit dynamic loading. Currently, the UMAT

developed for ABAQUS is only compatible with the implicit solver. This solver is

138

used to simulate the response of structures under static or implicit dynamic loading.

Modifications to the LAMPATNL code that interfaces with ABAQUS are required for

the user material to function in the explicit solver environment. Further expansion in

the dynamic modeling field necessitates the addition of rate dependent material

behavior. The current constitutive model used for LAMPATNL is strain dependent.

Additional work would be required to develop and validate a strain rate dependent

constitutive model.

The ability to model residual thermal stresses that resulted from

processing the composite materials was present in the LAM3D code. These thermal

modeling features were largely abandoned in the transition to nonlinear materials and

were similarly absent in the LAMPAT and LAMPATNL. With the framework of this

capability still in place, thermal modeling features could be reintroduced without much

difficultly. Strength predictions of metal matrix composite structures were shown to

be effected from initial thermal stresses in the structure [40]. Expanding the ability of

the user material to model thermal effects is a goal of future LAMPATNL

development.

As discussed in the Theoretical considerations section of Chapter 3, there

are changes to the procedure of calculating ply-level stresses and strains from

LAM3DNLP to LAMPATNL. In that section, it was noted that these changes do not

affect in-plane stress-strain predictions but may have implications for hybrid

composites and laminates exposed to out-of-plane loading. In order to implement

LAM3DNLP into the finite element code, it was required to not calculate the out-of-

plane ply strain in accordance with the original theory. Further research is required to

139

determine and discuss the effects, if any, of setting all ply strains equal to total

laminate strains.

140

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APPENDIX

REPRINT PERMISSION LETTER

To Whom It May Concern:

As the primary author of the papers “Three-dimensional effective property

and strength prediction of thick laminated composite media” published in Army

Research Laboratory Technical Report 911 and “Predicting the nonlinear response and

progressive failure of composite laminates” published in Composites Science and

Technology 64, I grant Jeffrey Staniszewski permission to use any figures contained in

these two reports.

Signed,

Dr. Travis A. Bogetti

U.S. Army Research Laboratory

302-831-6324